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TR11-028 | 24th February 2011
Richard Beigel, Bin Fu

A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects
a random element from the input list each time, we develop a
randomized $O({n(\log n)(\log\log n)\over ... more >>>


TR16-133 | 25th August 2016
Deeparnab Chakrabarty, C. Seshadhri

A $\widetilde{O}(n)$ Non-Adaptive Tester for Unateness

Revisions: 1

Khot and Shinkar (RANDOM, 2016) recently describe an adaptive, $O(n\log(n)/\varepsilon)$-query tester for unateness of Boolean functions $f:\{0,1\}^n \mapsto \{0,1\}$. In this note we describe a simple non-adaptive, $O(n\log(n/\varepsilon)/\varepsilon)$ -query tester for unateness for functions over the hypercube with any ordered range.

more >>>

TR07-047 | 15th May 2007
Dan Gutfreund, Alexander Healy, Tali Kaufman, Guy Rothblum

A (De)constructive Approach to Program Checking

Program checking, program self-correcting and program self-testing
were pioneered by [Blum and Kannan] and [Blum, Luby and Rubinfeld] in
the mid eighties as a new way to gain confidence in software, by
considering program correctness on an input by input basis rather than
full program verification. Work in ... more >>>


TR04-084 | 28th September 2004
George Karakostas

A better approximation ratio for the Vertex Cover problem

We reduce the approximation factor for Vertex Cover to $2-\Theta(1/\sqrt{logn})$
(instead of the previous $2-\Theta(loglogn/logn})$, obtained by Bar-Yehuda and Even,
and by Monien and Speckenmeyer in 1985. The improvement of the vanishing
factor comes as an application of the recent results of Arora, Rao, and Vazirani
that improved ... more >>>


TR15-166 | 17th October 2015
Magnus Gausdal Find, Alexander Golovnev, Edward Hirsch, Alexander Kulikov

A better-than-$3n$ lower bound for the circuit complexity of an explicit function

Revisions: 1

We consider Boolean circuits over the full binary basis. We prove a $(3+\frac{1}{86})n-o(n)$ lower bound on the size of such a circuit for an explicitly defined predicate, namely an affine disperser for sublinear dimension. This improves the $3n-o(n)$ bound of Norbert Blum (1984). The proof is based on the gate ... more >>>


TR09-028 | 2nd April 2009
Oded Goldreich

A Candidate Counterexample to the Easy Cylinders Conjecture

We present a candidate counterexample to the
easy cylinders conjecture, which was recently suggested
by Manindra Agrawal and Osamu Watanabe (ECCC, TR09-019).
Loosely speaking, the conjecture asserts that any 1-1 function
in $P/poly$ can be decomposed into ``cylinders'' of sub-exponential
size that can each be inverted by some polynomial-size circuit.
more >>>


TR11-021 | 13th February 2011
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Finding an efficient solution to the general problem of polynomial identity testing (PIT) is a challenging task. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-$3$ PIT, the other of depth-$4$ PIT.

Our first problem is ... more >>>


TR13-146 | 20th October 2013
Subhash Khot, Madhur Tulsiani, Pratik Worah

A Characterization of Approximation Resistance

Revisions: 1

A predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$ is called {\it approximation resistant} if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment that satisfies at least $\rho(f)+\Omega(1)$ fraction of the constraints.

We present a complete characterization of approximation resistant predicates under the ... more >>>


TR16-201 | 19th December 2016
Eric Blais, Yuichi Yoshida

A Characterization of Constant-Sample Testable Properties

We characterize the set of properties of Boolean-valued functions on a finite domain $\mathcal{X}$ that are testable with a constant number of samples.
Specifically, we show that a property $\mathcal{P}$ is testable with a constant number of samples if and only if it is (essentially) a $k$-part symmetric property ... more >>>


TR13-075 | 23rd May 2013
Subhash Khot, Madhur Tulsiani, Pratik Worah

A Characterization of Strong Approximation Resistance

For a predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$, we call the predicate strongly approximation resistant if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment such that the fraction of constraints satisfied is outside the range $[\rho(f)-\Omega(1), \rho(f)+\Omega(1)]$.

We present a characterization of ... more >>>


TR12-161 | 20th November 2012
Olaf Beyersdorff, Nicola Galesi, Massimo Lauria

A Characterization of Tree-Like Resolution Size

We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas and for the classical pigeonhole principle. The main point of this note is to show that the asymmetric game ... more >>>


TR14-156 | 26th November 2014
Jayadev Acharya, Clement Canonne, Gautam Kamath

A Chasm Between Identity and Equivalence Testing with Conditional Queries

Revisions: 1

A recent model for property testing of probability distributions enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain.
In particular, Canonne et al. showed that, in this setting, testing identity of an unknown distribution $D$ (i.e., ... more >>>


TR10-179 | 18th November 2010
Gregory Valiant, Paul Valiant

A CLT and tight lower bounds for estimating entropy

Revisions: 1

We prove two new multivariate central limit theorems; the first relates the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance matric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central ... more >>>


TR11-087 | 3rd June 2011
Michael Viderman

A Combination of Testability and Decodability by Tensor Products

Revisions: 3

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as ... more >>>


TR99-030 | 9th July 1999
Meena Mahajan, P R Subramanya, V. Vinay

A Combinatorial Algorithm for Pfaffians

The Pfaffian of an oriented graph is closely linked to
Perfect Matching. It is also naturally related to the determinant of
an appropriately defined matrix. This relation between Pfaffian and
determinant is usually exploited to give a fast algorithm for
computing Pfaffians.

We present the first completely combinatorial algorithm for ... more >>>


TR10-095 | 11th June 2010
Masaki Yamamoto

A combinatorial analysis for the critical clause tree

In [FOCS1998],
Paturi, Pudl\'ak, Saks, and Zane proposed a simple randomized algorithm
for finding a satisfying assignment of a $k$-CNF formula.
The main lemma of the paper is as follows:
Given a satisfiable $k$-CNF formula that
has a $d$-isolated satisfying assignment $z$,
the randomized algorithm finds $z$
with probability at ... more >>>


TR02-035 | 27th May 2002
Albert Atserias, Víctor Dalmau

A Combinatorial Characterization of Resolution Width

We provide a characterization of the resolution
width introduced in the context of Propositional Proof Complexity
in terms of the existential pebble game introduced
in the context of Finite Model Theory. The characterization
is tight and purely combinatorial. Our
first application of this result is a surprising
proof that the ... more >>>


TR12-159 | 20th November 2012
Eli Ben-Sasson, Michael Viderman

A Combinatorial Characterization of smooth LTCs and Applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs ... more >>>


TR03-044 | 12th May 2003
Juan Luis Esteban, Jacobo Toran

A Combinatorial Characterization of Treelike Resolution Space

We show that the Player-Adversary game from a paper
by Pudlak and Impagliazzo played over
CNF propositional formulas gives
an exact characterization of the space needed
in treelike resolution refutations. This
characterization is purely combinatorial
and independent of the notion of resolution.
We use this characterization to give ... more >>>


TR96-047 | 2nd September 1996
Oded Goldreich, Muli Safra

A Combinatorial Consistency Lemma with application to the PCP Theorem

Revisions: 1


The current proof of the PCP Theorem (i.e., NP=PCP(log,O(1)))
is very complicated.
One source of difficulty is the technically involved
analysis of low-degree tests.
Here, we refer to the difficulty of obtaining strong results
regarding low-degree tests; namely, results of the type obtained and
used by ... more >>>


TR07-066 | 24th April 2007
Maciej Liskiewicz, Christian Hundt

A Combinatorial Geometric Approach to Linear Image Matching

The problem of image matching is to find for two given digital images $A$ and $B$
an admissible transformation that converts image $A$ as close as possible to $B$.
This problem becomes hard if the space of admissible transformations is too complex.
Consequently, in many real applications, like the ones ... more >>>


TR96-039 | 27th June 1996
Carme Alvarez, Raymond Greenlaw

A Compendium of Problems Complete for Symmetric Logarithmic Space

Comments: 2

We provide a compendium of problems that are complete for
symmetric logarithmic space (SL). Complete problems are one method
of studying this class for which programming is nonintuitive. A
number of the problems in the list were not previously known to be
complete. A ... more >>>


TR10-018 | 15th February 2010
Vitaly Feldman

A Complete Characterization of Statistical Query Learning with Applications to Evolvability

Revisions: 1

Statistical query (SQ) learning model of Kearns (1993) is a natural restriction of the PAC learning model in which a learning algorithm is allowed to obtain estimates of statistical properties of the examples but cannot see the examples themselves. We describe a new and simple characterization of the query complexity ... more >>>


TR96-062 | 3rd December 1996
Sanjeev Khanna, Madhu Sudan, David P. Williamson

A Complete Characterization of the Approximability of Maximization Problems Derived from Boolean Constraint Satisfaction


In this paper we study the approximability of boolean constraint
satisfaction problems. A problem in this class consists of some
collection of ``constraints'' (i.e., functions
$f:\{0,1\}^k \rightarrow \{0,1\}$); an instance of a problem is a set
of constraints applied to specified subsets of $n$ boolean
variables. Schaefer earlier ... more >>>


TR00-084 | 6th November 2000
Salil Vadhan, Amit Sahai

A Complete Problem for Statistical Zero Knowledge

We present the first complete problem for SZK, the class of (promise)
problems possessing statistical zero-knowledge proofs (against an
honest verifier). The problem, called STATISTICAL DIFFERENCE, is to
decide whether two efficiently samplable distributions are either
statistically close or far apart. This gives a new characterization
of SZK that makes ... more >>>


TR06-046 | 1st April 2006
Dima Grigoriev, Edward Hirsch, Konstantin Pervyshev

A Complete Public-Key Cryptosystem

Comments: 1

We present a cryptosystem which is complete for the class of probabilistic public-key cryptosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal, this class contains probabilistic encryption of Goldwasser-Micali as well as Ajtai-Dwork and NTRU cryptosystems. The latter two are known to make errors with ... more >>>


TR10-092 | 22nd May 2010
Charanjit Jutla, Arnab Roy

A Completeness Theorem for Pseudo-Linear Functions with Applications to UC Security

Revisions: 1 , Comments: 1

We consider multivariate pseudo-linear functions
over finite fields of characteristic two. A pseudo-linear polynomial
is a sum of guarded linear-terms, where a guarded linear-term is a product of one or more linear-guards
and a single linear term, and each linear-guard is
again a linear term but raised ... more >>>


TR09-140 | 18th December 2009
Saugata Basu

A complex analogue of Toda's Theorem

Revisions: 1

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy,
$\mathbf{PH}$,
is contained in the class $\mathbf{P}^{\#\mathbf{P}}$,
namely the class of languages that can be
decided by a Turing machine in polynomial time given access to an
oracle with the power to compute a function in the ... more >>>


TR15-167 | 15th October 2015
Mika Göös, T.S. Jayram

A Composition Theorem for Conical Juntas

We describe a general method of proving degree lower bounds for conical juntas (nonnegative combinations of conjunctions) that compute recursively defined boolean functions. Such lower bounds are known to carry over to communication complexity. We give two applications:

$\bullet~$ $\textbf{AND-OR trees}$: We show a near-optimal $\tilde{\Omega}(n^{0.753...})$ randomised communication lower bound ... more >>>


TR96-059 | 12th November 1996
Shai Ben-David, Nader H. Bshouty, Eyal Kushilevitz

A Composition Theorem for Learning Algorithms with Applications to Geometric Concept Classes


This paper solves the open problem of exact learning
geometric objects bounded by hyperplanes (and more generally by any constant
degree algebraic surfaces) in the constant
dimensional space from equivalence queries only (i.e., in the on-line learning
model).

We present a novel approach that allows, under ... more >>>


TR15-142 | 28th August 2015
Srikanth Srinivasan

A Compression Algorithm for $AC^0[\oplus]$ circuits using Certifying Polynomials

A recent work of Chen, Kabanets, Kolokolova, Shaltiel and Zuckerman (CCC 2014, Computational Complexity 2015) introduced the Compression problem for a class $\mathcal{C}$ of circuits, defined as follows. Given as input the truth table of a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ that has a small (say size $s$) circuit from ... more >>>


TR08-046 | 14th April 2008
Nikhil R. Devanur, Lance Fortnow

A Computational Theory of Awareness and Decision Making

We exhibit a new computational-based definition of awareness,
informally that our level of unawareness of an object is the amount
of time needed to generate that object within a certain environment.
We give several examples to show this notion matches our intuition
in scenarios where one organizes, accesses and transfers
more >>>


TR11-051 | 8th April 2011
Thomas Vidick

A concentration inequality for the overlap of a vector on a large set, With application to the communication complexity of the Gap-Hamming-Distance problem

Given two sets $A,B\subseteq\R^n$, a measure of their dependence, or correlation, is given by the expected squared inner product between random $x\in A $ and $y\in B$. We prove an inequality showing that no two sets of large enough Gaussian measure (at least $e^{-\delta n}$ for some constant $\delta >0$) ... more >>>


TR02-061 | 14th November 2002
Miklos Ajtai

A conjectured 0-1 law about the polynomial time computable properties of random lattices, I.

A measure $\mu_{n}$ on $n$-dimensional lattices with
determinant $1$ was introduced about fifty years ago to prove the
existence of lattices which contain points from certain sets. $\mu_{n}$
is the unique probability measure on lattices with determinant $1$ which
is invariant under linear transformations with determinant $1$, where a
more >>>


TR98-010 | 22nd January 1998
Phong Nguyen, Jacques Stern

A Converse to the Ajtai-Dwork Security Proof and its Cryptographic Implications

Revisions: 1


Recently, Ajtai discovered a fascinating connection
between the worst-case complexity and the average-case
complexity of some well-known lattice problems.
Later, Ajtai and Dwork proposed a cryptosystem inspired
by Ajtai's work, provably secure if a particular lattice
problem is difficult. We show that there is a converse
to the ... more >>>


TR12-173 | 8th December 2012
Kfir Barhum, Thomas Holenstein

A Cookbook for Black-Box Separations and a Recipe for UOWHFs

We present a new framework for proving fully black-box
separations and lower bounds. We prove a general theorem that facilitates
the proofs of fully black-box lower bounds from a one-way function (OWF).

Loosely speaking, our theorem says that in order to prove that a fully black-box
construction does ... more >>>


TR08-018 | 28th February 2008
Ran Raz

A Counterexample to Strong Parallel Repetition

The parallel repetition theorem states that for any two-prover game,
with value $1- \epsilon$ (for, say, $\epsilon \leq 1/2$), the value of
the game repeated in parallel $n$ times is at most
$(1- \epsilon^c)^{\Omega(n/s)}$, where $s$ is the answers' length
(of the original game) and $c$ is a universal ... more >>>


TR10-026 | 25th February 2010
Hao Huang, Benny Sudakov

A counterexample to the Alon-Saks-Seymour conjecture and related problems

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs.
Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$.
This well known conjecture remained open for almost twenty years.
In this paper, we construct a counterexample to this
conjecture and discuss ... more >>>


TR10-109 | 11th July 2010
Scott Aaronson

A Counterexample to the Generalized Linial-Nisan Conjecture

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth ... more >>>


TR97-055 | 22nd September 1997
Bruce Edward Litow

A Decision Method for the Rational Sequence Problem

We give a method to decide whether or not an
ordinary finite order linear recurrence with constant, rational
coefficients ever generates zero.

more >>>

TR10-098 | 17th June 2010
Daniel Kane, Jelani Nelson

A Derandomized Sparse Johnson-Lindenstrauss Transform

Revisions: 2

Recent work of [Dasgupta-Kumar-Sarl\'{o}s, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative proof of their result requiring only bounded independence hash functions. Furthermore, the sparsity bound obtained in ... more >>>


TR12-116 | 13th September 2012
Luca Trevisan

A Derandomized Switching Lemma and an Improved Derandomization of AC0

Revisions: 1

We describe a new pseudorandom generator for AC0. Our generator $\epsilon$-fools circuits of depth $d$ and size $M$ and uses a seed of length $\tilde O( \log^{d+4} M/\epsilon)$. The previous best construction for $d \geq 3$ was due to Nisan, and had seed length $O(\log^{2d+6} M/\epsilon)$.
A seed length of ... more >>>


TR10-014 | 2nd February 2010
Daniele Micciancio, Panagiotis Voulgaris

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations

Revisions: 1

We give deterministic $2^{O(n)}$-time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP).
This improves the $n^{O(n)}$ running time of the best previously known algorithms for CVP (Kannan, ... more >>>


TR11-126 | 17th September 2011
Benny Applebaum, Andrej Bogdanov, Alon Rosen

A Dichotomy for Local Small-Bias Generators

We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: they can be described by a sparse input-output dependency graph and a small predicate that is applied at each output. Following the works of Cryan and Miltersen ... more >>>


TR07-029 | 20th January 2007
Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto

A Dichotomy Theorem within Schaefer for the Boolean Connectivity Problem

Revisions: 1

P. Gopalan, P. G. Kolaitis, E. N. Maneva and C. H. Papadimitriou
studied in [Gopalan et al., ICALP2006] connectivity properties of the
solution-space of Boolean formulas, and investigated complexity issues
on connectivity problems in Schaefer's framework [Schaefer, STOC1978].
A set S of logical relations is Schaefer if all relations in ... more >>>


TR11-164 | 9th December 2011
Mark Braverman, Omri Weinstein

A discrepancy lower bound for information complexity

This paper provides the first general technique for proving information lower bounds on two-party
unbounded-rounds communication problems. We show that the discrepancy lower bound, which
applies to randomized communication complexity, also applies to information complexity. More
precisely, if the discrepancy of a two-party function $f$ with respect ... more >>>


TR98-050 | 6th July 1998
Farid Ablayev, Svetlana Ablayeva

A Discrete Approximation and Communication Complexity Approach to the Superposition Problem

The superposition (or composition) problem is a problem of
representation of a function $f$ by a superposition of "simpler" (in a
different meanings) set $\Omega$ of functions. In terms of circuits
theory this means a possibility of computing $f$ by a finite circuit
with 1 fan-out gates $\Omega$ of functions. ... more >>>


TR08-106 | 12th November 2008
Jack H. Lutz

A Divergence Formula for Randomness and Dimension

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $\beta$ is ... more >>>


TR09-037 | 10th April 2009
Parikshit Gopalan

A Fourier-analytic approach to Reed-Muller decoding

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We prove that the list-decoding radius for quadratic polynomials equals $1 - 2/q$ over any field $F_q$ where $q > 2$. This confirms a conjecture due to Gopalan, Klivans and Zuckerman for degree $2$. Previously, tight bounds ... more >>>


TR13-027 | 29th January 2013
Luke Friedman

A Framework for Proving Proof Complexity Lower Bounds on Random CNFs Using Encoding Techniques

Propositional proof complexity is an area of complexity theory that addresses the question of whether the class NP is closed under complement, and also provides a theoretical framework for studying practical applications such as SAT solving.
Some of the most well-studied contradictions are random $k$-CNF formulas where each clause of ... more >>>


TR10-057 | 1st April 2010
Scott Aaronson, Andrew Drucker

A Full Characterization of Quantum Advice

Revisions: 3

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. ... more >>>


TR11-036 | 17th March 2011
Gilad Asharov, Yehuda Lindell

A Full Proof of the BGW Protocol for Perfectly-Secure Multiparty Computation

Revisions: 4

In the setting of secure multiparty computation, a set of $n$ parties with private inputs wish to jointly compute some functionality of their inputs. One of the most fundamental results of information-theoretically secure computation was presented by Ben-Or, Goldwasser and Wigderson (BGW) in 1988. They demonstrated that any $n$-party functionality ... more >>>


TR14-131 | 7th October 2014
Olaf Beyersdorff, Leroy Chew, Karteek Sreenivasaiah

A game characterisation of tree-like Q-Resolution size

We provide a characterisation for the size of proofs in tree-like Q-Resolution by a Prover-Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives the first successful transfer of one of the lower bound techniques for classical proof systems to QBF ... more >>>


TR02-003 | 24th December 2001
Eli Ben-Sasson, Yonatan Bilu

A Gap in Average Proof Complexity

We present the first example of a natural distribution on instances
of an NP-complete problem, with the following properties.
With high probability a random formula from this
distribution (a) is unsatisfiable,
(b) has a short proof that can be found easily, and (c) does not have a short
(general) resolution ... more >>>


TR02-058 | 25th September 2002
Philippe Moser

A generalization of Lutz's measure to probabilistic classes

We extend Lutz's measure to probabilistic classes, and obtain notions of measure on probabilistic complexity classes
C
such as BPP , BPE and BPEXP. Unlike former attempts,
all our measure notions satisfy all three Lutz's measure axioms, that is
every singleton {L} has measure zero ... more >>>


TR98-058 | 2nd August 1998
H. Buhrman, Dieter van Melkebeek, K.W. Regan, Martin Strauss, D. Sivakumar

A Generalization of Resource-Bounded Measure, With Application to the BPP vs. EXP Problem

We introduce "resource-bounded betting games", and propose
a generalization of Lutz's resource-bounded measure in which the choice
of next string to bet on is fully adaptive. Lutz's martingales are
equivalent to betting games constrained to bet on strings in lexicographic
order. We show that if strong pseudo-random number generators exist,
more >>>


TR13-093 | 21st June 2013
Anna Gal, Jing-Tang Jang

A Generalization of Spira's Theorem and Circuits with Small Segregators or Separators

Spira showed that any Boolean formula of size $s$ can be simulated in depth $O(\log s)$. We generalize Spira's theorem and show that any Boolean circuit of size $s$ with segregators of size $f(s)$ can be simulated in depth $O(f(s)\log s)$. If the segregator size is at least $s^{\varepsilon}$ for ... more >>>


TR15-078 | 4th May 2015
Mladen Mikša, Jakob Nordström

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good ... more >>>


TR05-111 | 3rd October 2005
Dieter van Melkebeek, Konstantin Pervyshev

A Generic Time Hierarchy for Semantic Models With One Bit of Advice

We show that for any reasonable semantic model of computation and for
any positive integer $a$ and rationals $1 \leq c < d$, there exists a language
computable in time $n^d$ with $a$ bits of advice but not in time $n^c$
with $a$ bits of advice. A semantic ... more >>>


TR05-009 | 14th December 2004
David P. Woodruff, Sergey Yekhanin

A Geometric Approach to Information-Theoretic Private Information Retrieval

A t-private private information retrieval (PIR) scheme allows a user to retrieve the i-th bit of an n-bit string x replicated among k servers, while any coalition of up to t servers learns no information about i. We present a new geometric approach to PIR, and obtain (1) A t-private ... more >>>


TR09-096 | 7th October 2009
Haralampos Tsaknakis, Paul Spirakis

A Graph Spectral Approach for Computing Approximate Nash Equilibria

We present a new methodology for computing approximate Nash equilibria for two-person non-cooperative games based
upon certain extensions and specializations of an existing optimization approach previously used for the derivation of fixed approximations for this problem. In particular, the general two-person problem is reduced to an indefinite quadratic programming problem ... more >>>


TR15-115 | 20th July 2015
Ilya Volkovich

A Guide to Learning Arithmetic Circuits

An \emph{arithmetic circuit} is a directed acyclic graph in which the operations are $\{+,\times\}$.
In this paper, we exhibit several connections between learning algorithms for arithmetic circuits and other problems.
In particular, we show that:

\begin{enumerate}
\item Efficient learning algorithms for arithmetic circuit classes imply explicit exponential lower bounds.

... more >>>

TR96-050 | 23rd September 1996
Petr Savicky, Stanislav Zak

A hierarchy for (1,+k)-branching programs with respect to k

Branching programs (b.p.'s) or decision diagrams are a general
graph-based model of sequential computation. The b.p.'s of
polynomial size are a nonuniform counterpart of LOG. Lower bounds
for different kinds of restricted b.p.'s are intensively
investigated. An important restriction are so called $k$-b.p.'s,
where each computation reads each input ... more >>>


TR01-017 | 14th February 2001
Petr Savicky, Detlef Sieling

A Hierarchy Result for Read-Once Branching Programs with Restricted Parity Nondeterminism

Restricted branching programs are considered in complexity theory in
order to study the space complexity of sequential computations and
in applications as a data structure for Boolean functions. In this
paper (\oplus,k)-branching programs and (\vee,k)-branching
programs are considered, i.e., branching programs starting with a
... more >>>


TR08-098 | 12th November 2008
Victor Chen

A Hypergraph Dictatorship Test with Perfect Completeness

Revisions: 1

A hypergraph dictatorship test is first introduced by Samorodnitsky
and Trevisan and serves as a key component in
their unique games based $\PCP$ construction. Such a test has oracle
access to a collection of functions and determines whether all the
functions are the same dictatorship, or all their low degree ... more >>>


TR01-059 | 20th July 2001
Elvira Mayordomo

A Kolmogorov complexity characterization of constructive Hausdorff dimension

Revisions: 3

We obtain the following full characterization of constructive dimension
in terms of algorithmic information content. For every sequence A,
cdim(A)=liminf_n (K(A[0..n-1])/n.

more >>>

TR11-018 | 8th February 2011
Jochen Messner, Thomas Thierauf

A Kolmogorov Complexity Proof of the Lovász Local Lemma for Satisfiability

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

more >>>

TR96-036 | 28th May 1996
Petr Savicky, Stanislav Zak

A large lower bound for 1-branching programs

Revisions: 1

Branching programs (b.p.'s) or decision diagrams are a general
graph-based model of sequential computation. B.p.'s of polynomial
size are a nonuniform counterpart of LOG. Lower bounds for
different kinds of restricted b.p.'s are intensively investigated.
An important restriction are so called 1-b.p.'s, where each
computation reads each input bit at ... more >>>


TR06-098 | 17th August 2006
Grant Schoenebeck, Luca Trevisan, Madhur Tulsiani

A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover

We study semidefinite programming relaxations of Vertex Cover arising from
repeated applications of the LS+ ``lift-and-project'' method of Lovasz and
Schrijver starting from the standard linear programming relaxation.

Goemans and Kleinberg prove that after one round of LS+ the integrality
gap remains arbitrarily close to 2. Charikar proves an integrality ... more >>>


TR00-074 | 12th July 2000
Daniele Micciancio, Bogdan Warinschi

A Linear Space Algorithm for Computing the Hermite Normal Form

Computing the Hermite Normal Form
of an $n\times n$ matrix using the best current algorithms typically
requires $O(n^3\log M)$ space, where $M$ is a bound on the length of
the columns of the input matrix.
Although polynomial in the input size (which ... more >>>


TR11-043 | 25th March 2011
Scott Aaronson

A Linear-Optical Proof that the Permanent is #P-Hard

One of the crown jewels of complexity theory is Valiant's 1979 theorem that computing the permanent of an n*n matrix is #P-hard. Here we show that, by using the model of linear-optical quantum computing---and in particular, a universality theorem due to Knill, Laflamme, and Milburn---one can give a different and ... more >>>


TR09-049 | 5th May 2009
Derrick Stolee, Derrick Stolee, Chris Bourke, Vinodchandran Variyam

A log-space algorithm for reachability in planar DAGs with few sources

Designing algorithms that use logarithmic space for graph reachability problems is fundamental to complexity theory. It is well known that for general directed graphs this problem is equivalent to the NL vs L problem. For planar graphs, the question is not settled. Showing that the planar reachability problem is NL-complete ... more >>>


TR08-083 | 10th June 2008
Yijia Chen, Jörg Flum

A logic for PTIME and a parameterized halting problem

In [Blass, Gurevich, and Shelah, 99] a logic L_Y has been introduced as a possible candidate for a logic capturing the PTIME properties of structures (even in the absence of an ordering of their universe). A reformulation of this problem in terms of a parameterized halting problem p-Acc for nondeterministic ... more >>>


TR10-087 | 17th May 2010
Shachar Lovett, Ely Porat

A lower bound for dynamic approximate membership data structures

An approximate membership data structure is a randomized data
structure for representing a set which supports membership
queries. It allows for a small false positive error rate but has
no false negative errors. Such data structures were first
introduced by Bloom in the 1970's, and have since had numerous
applications, ... more >>>


TR98-011 | 29th January 1998
Farid Ablayev, Marek Karpinski

A Lower Bound for Integer Multiplication on Randomized Read-Once Branching Programs

We prove an exponential lower bound ($2^{\Omega(n/\log n)}$) on the
size of any randomized ordered read-once branching program
computing integer multiplication. Our proof depends on proving
a new lower bound on Yao's randomized one-way communication
complexity of certain boolean functions. It generalizes to some
other ... more >>>


TR99-010 | 1st April 1999
Eric Allender, Igor E. Shparlinski, Michael Saks

A Lower Bound for Primality

Comments: 1

Recent work by Bernasconi, Damm and Shparlinski
proved lower bounds on the circuit complexity of the square-free
numbers, and raised as an open question if similar (or stronger)
lower bounds could be proved for the set of prime numbers. In
this short note, we answer this question ... more >>>


TR95-063 | 19th December 1995
Dima Grigoriev, Marek Karpinski, Friedhelm Meyer auf der Heide, Roman Smolensky

A Lower Bound for Randomized Algebraic Decision Trees

We extend the lower bounds on the depth of algebraic decision trees
to the case of {\em randomized} algebraic decision trees (with
two-sided error) for languages being finite unions of hyperplanes
and the intersections of halfspaces, solving a long standing open
problem. As an application, among ... more >>>


TR97-019 | 5th May 1997
Martin Sauerhoff

A Lower Bound for Randomized Read-k-Times Branching Programs

In this paper, we are concerned with randomized OBDDs and randomized
read-k-times branching programs. We present an example of a Boolean
function which has polynomial size randomized OBDDs with small,
one-sided error, but only non-deterministic read-once branching
programs of exponential size. Furthermore, we discuss a lower bound
technique for randomized ... more >>>


TR10-081 | 10th May 2010
Olaf Beyersdorff, Nicola Galesi, Massimo Lauria

A Lower Bound for the Pigeonhole Principle in Tree-like Resolution by Asymmetric Prover-Delayer Games

In this note we show that the asymmetric Prover-Delayer game developed by Beyersdorff, Galesi, and Lauria (ECCC TR10-059) for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form $2^{\Omega(n\log n)}$ for the pigeonhole ... more >>>


TR06-060 | 4th May 2006
Ran Raz, Amir Shpilka, Amir Yehudayoff

A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits

We construct an explicit polynomial $f(x_1,...,x_n)$, with
coefficients in ${0,1}$, such that the size of any syntactically
multilinear arithmetic circuit computing $f$ is at least
$\Omega( n^{4/3} / log^2(n) )$. The lower bound holds over any field.

more >>>

TR02-002 | 3rd January 2002
Howard Barnum, Michael Saks

A lower bound on the quantum query complexity of read-once functions

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions.Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' ... more >>>


TR11-162 | 7th December 2011
Pavel Pudlak

A lower bound on the size of resolution proofs of the Ramsey theorem

We prove an exponential lower bound on the lengths of resolution proofs of propositions expressing the finite Ramsey theorem for pairs.

more >>>

TR08-110 | 19th November 2008
Chris Calabro

A Lower Bound on the Size of Series-Parallel Graphs Dense in Long Paths

One way to quantify how dense a multidag is in long paths is to find
the largest n, m such that whichever &le; n edges are removed, there is still
a path from an original input to an original output with &ge; m edges
- the larger ... more >>>


TR01-101 | 14th December 2001
Philipp Woelfel

A Lower Bound Technique for Restricted Branching Programs and Applications

We present a new lower bound technique for two types of restricted
Branching Programs (BPs), namely for read-once BPs (BP1s) with
restricted amount of nondeterminism and for (1,+k)-BPs. For this
technique, we introduce the notion of (strictly) k-wise l-mixed
Boolean functions, which generalizes the concept of l-mixedness ... more >>>


TR12-085 | 5th July 2012
Tsuyoshi Ito, Thomas Vidick

A multi-prover interactive proof for NEXP sound against entangled provers

We prove a strong limitation on the ability of entangled provers to collude in a multiplayer game. Our main result is the first nontrivial lower bound on the class MIP* of languages having multi-prover interactive proofs with entangled provers; namely MIP* contains NEXP, the class of languages decidable in non-deterministic ... more >>>


TR09-015 | 19th February 2009
Joshua Brody, Amit Chakrabarti

A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences

The Gap-Hamming-Distance problem arose in the context of proving space
lower bounds for a number of key problems in the data stream model. In
this problem, Alice and Bob have to decide whether the Hamming distance
between their $n$-bit input strings is large (i.e., at least $n/2 +
\sqrt n$) ... more >>>


TR17-051 | 16th March 2017
Mark Bun, Justin Thaler

A Nearly Optimal Lower Bound on the Approximate Degree of AC$^0$

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by ... more >>>


TR03-087 | 10th December 2003
Richard Beigel, Lance Fortnow, William Gasarch

A Nearly Tight Bound for Private Information Retrieval Protocols

Comments: 1

We show that any 1-round 2-server Private Information
Retrieval Protocol where the answers are 1-bit long must ask questions
that are at least $n-2$ bits long, which is nearly equal to the known
$n-1$ upper bound. This improves upon the approximately $0.25n$ lower
bound of Kerenidis and de Wolf while ... more >>>


TR16-058 | 12th April 2016
Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh Kothari, Ankur Moitra, Aaron Potechin

A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$.
This yields a nearly tight ... more >>>


TR99-036 | 6th September 1999
Edward Hirsch

A New Algorithm for MAX-2-SAT

Revisions: 2

Recently there was a significant progress in
proving (exponential-time) worst-case upper bounds for the
propositional satisfiability problem (SAT).
MAX-SAT is an important generalization of SAT.
Several upper bounds were obtained for MAX-SAT and
its NP-complete subproblems.
In particular, Niedermeier and Rossmanith recently
... more >>>


TR04-032 | 5th February 2004
Ryan Williams

A new algorithm for optimal constraint satisfaction and its implications

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a constant factor improvement in the base of ... more >>>


TR16-074 | 9th May 2016
Ilias Diakonikolas, Daniel Kane

A New Approach for Testing Properties of Discrete Distributions

We study problems in distribution property testing:
Given sample access to one or more unknown discrete distributions,
we want to determine whether they have some global property or are $\epsilon$-far
from having the property in $\ell_1$ distance (equivalently, total variation distance, or ``statistical distance'').
In this work, we give a ... more >>>


TR10-064 | 13th April 2010
Xin Li

A New Approach to Affine Extractors and Dispersers

We study the problem of constructing affine extractors over $\mathsf{GF(2)}$. Previously the only known construction that can handle sources with arbitrarily linear entropy is due to Bourgain (and a slight modification by Yehudayoff), which relies heavily on the technique of Van der Corput differencing and a careful choice of a ... more >>>


TR15-161 | 24th September 2015
Chaoping Xing, chen yuan

A new class of rank-metric codes and their list decoding beyond the unique decoding radius

Compared with classical block codes, efficient list decoding of rank-metric codes seems more difficult. The evidences to support this view include: (i) so far people have not found polynomial time list decoding algorithms of rank-metric codes with decoding radius beyond $(1-R)/2$ (where $R$ is the rate of code) if ratio ... more >>>


TR98-013 | 3rd March 1998
Nader H. Bshouty

A New Composition Theorem for Learning Algorithms


We present a new approach to the composition
of learning algorithms (in various models) for
classes of constant VC-dimension into learning algorithms for
more complicated classes.
We prove that if a class $\CC$ is learnable
in time $t$ from a hypothesis class $\HH$ of constant VC-dimension
then the class ... more >>>


TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>


TR06-096 | 10th August 2006
Iftach Haitner, Omer Reingold

A New Interactive Hashing Theorem

Interactive hashing, introduced by Naor et al. [NOVY98], plays
an important role in many cryptographic protocols. In particular, it
is a major component in all known constructions of
statistically-hiding commitment schemes and of zero-knowledge
arguments based on general one-way permutations and on one-way
functions. Interactive hashing with respect to a ... more >>>


TR09-059 | 2nd July 2009
Gábor Kun, Mario Szegedy

A NEW LINE OF ATTACK ON THE DICHOTOMY CONJECTURE

The well known dichotomy conjecture of Feder and
Vardi states that for every &#64257;nite family &#915; of constraints CSP(&#915;) is
either polynomially solvable or NP-hard. Bulatov and Jeavons re-
formulated this conjecture in terms of the properties of the algebra
P ol(&#915;), where the latter is ... more >>>


TR09-025 | 11th March 2009
Arnaldo Moura, Igor Carboni Oliveira

A New Look at Some Classical Results in Computational Complexity

We propose a generalization of the traditional algorithmic space and
time complexities. Using the concept introduced, we derive an
unified proof for the deterministic time and space hierarchy
theorems, now stated in a much more general setting. This opens the
possibility for the unification and generalization of other results
that ... more >>>


TR10-001 | 30th December 2009
Iftach Haitner, Mohammad Mahmoody, David Xiao

A New Sampling Protocol and Applications to Basing Cryptographic Primitives on the Hardness of $NP$

We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, $Sam_d$ where $d$ is the recursion depth, is based on ... more >>>


TR98-005 | 27th January 1998
Jin-Yi Cai

A new transference theorem and applications to Ajtai's connection factor

We prove a new transference theorem in the geometry of numbers,
giving optimal bounds relating the successive minima of a lattice
with the minimal length of generating vectors of its dual.
It generalizes the transference theorem due to Banaszczyk.
We also prove a stronger bound for the special class of ... more >>>


TR12-046 | 24th April 2012
Madhu Sudan, Noga Ron-Zewi

A new upper bound on the query complexity for testing generalized Reed-Muller codes

Revisions: 1

Over a finite field $\F_q$ the $(n,d,q)$-Reed-Muller code is the code given by evaluations of $n$-variate polynomials of total degree at most $d$ on all points (of $\F_q^n$). The task of testing if a function $f:\F_q^n \to \F_q$ is close to a codeword of an $(n,d,q)$-Reed-Muller code has been of ... more >>>


TR99-026 | 7th July 1999
Miklos Ajtai

A Non-linear Time Lower Bound for Boolean Branching Programs

We prove that for all positive integer $k$ and for all
sufficiently small $\epsilon >0$ if $n$ is sufficiently large
then there is no Boolean (or $2$-way) branching program of size
less than $2^{\epsilon n}$ which for all inputs
$X\subseteq \lbrace 0,1,...,n-1\rbrace $ computes in time $kn$
the parity of ... more >>>


TR05-122 | 31st October 2005
Pavel Pudlak

A nonlinear bound on the number of wires in bounded depth circuits

We shall prove a lower bound on the number of edges in some bounded
depth graphs. This theorem is stronger than lower bounds proved on
bounded depth superconcentrators and enables us to prove a lower bound
on certain bounded depth circuits for which we cannot use
superconcentrators: we prove that ... more >>>


TR06-089 | 16th July 2006
Sofya Raskhodnikova, Adam Smith

A Note on Adaptivity in Testing Properties of Bounded Degree Graphs

We show that in the bounded degree model for graph property testing,
adaptivity is essential. An algorithm is *non-adaptive* if it makes all queries to the input before receiving any answers. We call a property *non-trivial* if it does not depend only on the degree distribution of the nodes. We ... more >>>


TR10-134 | 23rd August 2010
Avraham Ben-Aroya, Klim Efremenko, Amnon Ta-Shma

A Note on Amplifying the Error-Tolerance of Locally Decodable Codes

Revisions: 2

We show a generic, simple way to amplify the error-tolerance of locally decodable codes.
Specifically, we show how to transform a locally decodable code that can tolerate a constant fraction of errors
to a locally decodable code that can recover from a much higher error-rate. We also show how to ... more >>>


TR00-043 | 21st June 2000
Uriel Feige, Marek Karpinski, Michael Langberg

A Note on Approximating MAX-BISECTION on Regular Graphs


We design a $0.795$ approximation algorithm for the Max-Bisection problem
restricted to regular graphs. In the case of three regular graphs our
results imply an approximation ratio of $0.834$.

more >>>

TR13-106 | 29th July 2013
Shay Moran, Amir Yehudayoff

A note on average-case sorting

Revisions: 2

This note studies the average-case comparison-complexity of sorting n elements when there is a known distribution on inputs and the goal is to minimize
the expected number of comparisons. We generalize Fredman's algorithm which
is a variant of insertion sort and provide a basically tight upper bound: If \mu is
more >>>


TR10-099 | 20th June 2010
T.C. Vijayaraghavan

A Note on Closure Properties of ModL

Recently in [Vij09, Corollary 3.7] the complexity class ModL has been shown to be closed under complement assuming NL = UL. In this note we continue to show many other closure properties of ModL which include the following.

1. ModL is closed under $\leq ^L_m$ reduction, $\vee$(join) and $\leq ^{UL}_m$ ... more >>>


TR02-069 | 14th November 2002
Luca Trevisan

A Note on Deterministic Approximate Counting for k-DNF

Revisions: 1

We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to ... more >>>


TR04-047 | 22nd April 2004
Xiaoyang Gu

A note on dimensions of polynomial size circuits

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension $0$. We also show that $\Ppoly^\io$ has $\pthree$-dimension $1/2$, $\pthree$-strong dimension $1$. Our results improve previous measure results of Lutz (1992) and dimension ... more >>>


TR09-069 | 2nd September 2009
Parikshit Gopalan

A note on Efremenko's Locally Decodable Codes

Revisions: 1

Building on work of Yekhanin and Raghavendra, Efremenko recently gave an elegant construction of 3-query LDCs which achieve sub-exponential length unconditionally.In this note, we observe that this construction can be viewed in the framework of Reed-Muller codes.

more >>>

TR10-174 | 12th November 2010
Scott Aaronson, Baris Aydinlioglu, Harry Buhrman, John Hitchcock, Dieter van Melkebeek

A note on exponential circuit lower bounds from derandomizing Arthur-Merlin games

We present an alternate proof of the recent result by Gutfreund and Kawachi that derandomizing Arthur-Merlin games into $P^{NP}$ implies linear-exponential circuit lower bounds for $E^{NP}$. Our proof is simpler and yields stronger results. In particular, consider the promise-$AM$ problem of distinguishing between the case where a given Boolean circuit ... more >>>


TR10-171 | 11th November 2010
Michael Viderman

A Note on high-rate Locally Testable Codes with sublinear query complexity

Inspired by recent construction of high-rate locally correctable codes with sublinear query complexity due to
Kopparty, Saraf and Yekhanin (2010) we address the similar question for locally testable codes (LTCs).

In this note we show a construction of high-rate LTCs with sublinear query complexity.
More formally, we show that for ... more >>>


TR99-009 | 26th March 1999
Marek Karpinski, Rustam Mubarakzjanov

A Note on Las Vegas OBDDs

We prove that the error-free (Las Vegas) randomized OBDDs
are computationally equivalent to the deterministic OBDDs.
In contrast, it is known the same is not true for the
Las Vegas read-once branching programs.

more >>>

TR17-048 | 14th March 2017
Pavel Hrubes, Pavel Pudlak

A note on monotone real circuits

We show that if a Boolean function $f:\{0,1\}^n\to \{0,1\}$ can be computed by a monotone real circuit of size $s$ using $k$-ary monotone gates then $f$ can be computed by a monotone real circuit of size $O(sn^{k-2})$ which uses unary or binary monotone gates only. This partially solves an open ... more >>>


TR15-187 | 24th November 2015
Nir Bitansky, Vinod Vaikuntanathan

A Note on Perfect Correctness by Derandomization

Revisions: 1


In this note, we show how to transform a large class of erroneous cryptographic schemes into perfectly correct ones. The transformation works for schemes that are correct on every input with probability noticeably larger than half, and are secure under parallel repetition. We assume the existence of one-way functions ... more >>>


TR10-100 | 25th June 2010
Amit Chakrabarti

A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem

The deterministic space complexity of approximating the length of the longest increasing subsequence of
a stream of $N$ integers is known to be $\widetilde{\Theta}(\sqrt N)$. However, the randomized
complexity is wide open. We show that the technique used in earlier work to establish the $\Omega(\sqrt
N)$ deterministic lower bound fails ... more >>>


TR94-027 | 12th December 1994
Stasys Jukna

A Note on Read-k Times Branching Programs

A syntactic read-k times branching program has the restriction
that no variable occurs more than k times on any path (whether or not
consistent). We exhibit an explicit Boolean function f which cannot
be computed by nondeterministic syntactic read-k times branching programs
of size less than exp(\sqrt{n}}k^{-2k}), ... more >>>


TR95-009 | 2nd February 1995
Matthias Krause

A note on realizing iterated multiplication by small depth threshold circuits

It is shown that decomposition via Chinise Remainder does not
yield polynomial size depth 3 threshold circuits for iterated
multiplication of n-bit numbers. This result is achieved by
proving that, in contrast to multiplication of two n-bit
numbers, powering, division, and other related problems, the
... more >>>


TR13-128 | 16th September 2013
Pavel Hrubes

A note on semantic cutting planes

We show that the semantic cutting planes proof system has feasible interpolation via monotone real circuits. This gives an exponential lower bound on proof length in the system.

We also pose the following problem: can every multivariate non-decreasing function be expressed as a composition of non-decreasing functions in two ... more >>>


TR07-105 | 21st September 2007
Jelani Nelson

A Note on Set Cover Inapproximability Independent of Universe Size

Revisions: 1

In the set cover problem we are given a collection of $m$ sets whose union covers $[n] = \{1,\ldots,n\}$ and must find a minimum-sized subcollection whose union still covers $[n]$. We investigate the approximability of set cover by an approximation ratio that depends only on $m$ and observe that, for ... more >>>


TR01-029 | 27th March 2001
Denis Xavier Charles

A Note on Subgroup Membership Problem for PSL(2,p).

Comments: 1

We show that there are infinitely many primes $p$, such
that the subgroup membership problem for PSL(2,p) belongs
to $\NP \cap \coNP$.

more >>>

TR12-095 | 23rd July 2012
Avraham Ben-Aroya, Igor Shinkar

A Note on Subspace Evasive Sets

A subspace-evasive set over a field ${\mathbb F}$ is a subset of ${\mathbb F}^n$ that has small intersection with any low-dimensional affine subspace of ${\mathbb F}^n$. Interest in subspace evasive sets began in the work of Pudlák and Rödl (Quaderni di Matematica 2004). More recently, Guruswami (CCC 2011) showed that ... more >>>


TR16-065 | 18th April 2016
Xi Chen, Yu Cheng, Bo Tang

A Note on Teaching for VC Classes

Revisions: 1

In this note, we study the recursive teaching dimension(RTD) of concept classes of low VC-dimension. Recall that the VC-dimension of $C \subseteq \{0,1\}^n$, denoted by $VCD(C)$, is the maximum size of a shattered subset of $[n]$, where $Y\subseteq [n]$ is shattered if for every binary string $\vec{b}$ of length $|Y|$, ... more >>>


TR05-130 | 31st October 2005
Ahuva Mu'alem

A Note on Testing Truthfulness

This work initiates the study of algorithms
for the testing of monotonicity of mechanisms.
Such testing algorithms are useful for
searching dominant strategy mechanisms.
An $\e$-tester for monotonicity
is given a query access to a mechanism,
accepts if monotonicity is satisfied,
and rejects with high probability if more than $\e$-fraction
more >>>


TR04-056 | 1st July 2004
Vinodchandran Variyam

A note on the circuit complexity of PP

In this short note we show that for any integer k, there are
languages in the complexity class PP that do not have Boolean
circuits of size $n^k$.

more >>>

TR13-069 | 1st May 2013
Kousha Etessami, Alistair Stewart, Mihalis Yannakakis

A note on the complexity of comparing succinctly represented integers, with an application to maximum probability parsing

The following two decision problems capture the complexity of
comparing integers or rationals that are succinctly represented in
product-of-exponentials notation, or equivalently, via arithmetic
circuits using only multiplication and division gates, and integer
inputs:

Input instance: four lists of positive integers:

$a_1, \ldots , a_n; \ b_1, \ldots ,b_n; \ ... more >>>


TR01-092 | 2nd October 2001
Till Tantau

A Note on the Complexity of the Reachability Problem for Tournaments

Deciding whether a vertex in a graph is reachable from another
vertex has been studied intensively in complexity theory and is
well understood. For common types of graphs like directed graphs,
undirected graphs, dags or trees it takes a (possibly
nondeterministic) logspace machine to decide the reachability
problem, and ... more >>>


TR06-076 | 4th May 2006
Noam Nisan

A Note on the computational hardness of evolutionary stable strategies

We present a very simple reduction that when given a graph G and an integer k produces a game that has an evolutionary stable strategy if and only if the maximum clique size of G is not exactly k. Formally this shows that existence of evolutionary stable strategies is hard ... more >>>


TR08-012 | 20th November 2007
Arnab Bhattacharyya

A Note on the Distance to Monotonicity of Boolean Functions

Given a boolean function, let epsilon_M(f) denote the smallest distance between f and a monotone function on {0,1}^n. Let delta_M(f) denote the fraction of hypercube edges where f violates monotonicity. We give an alternative proof of the tight bound: delta_M(f) >= 2/n eps_M(f) for any boolean function f. This was ... more >>>


TR00-088 | 28th November 2000
Meena Mahajan, V. Vinay

A note on the hardness of the characteristic polynomial


In this note, we consider the problem of computing the
coefficients of the characteristic polynomial of a given
matrix, and the related problem of verifying the
coefficents.

Santha and Tan [CC98] show that verifying the determinant
(the constant term in the characteristic polynomial) is
complete for the class C=L, ... more >>>


TR12-092 | 6th July 2012
Pavol Duris

A Note On the Hierarchy of One-way Data-Independent Multi-Head Finite Automata.

In this paper we deal with one-way multi-head data-independent finite automata. A $k$-head finite automaton $A$ is data-independent, if the position of every head $i$ after step $t$ in the computation on an input $w$ is a function that depends only on the length of the input $w$, on $i$ ... more >>>


TR01-058 | 28th August 2001
Stasys Jukna

A Note on the Minimum Number of Negations Leading to Superpolynomial Savings

In 1957 Markov proved that every circuit in $n$ variables
can be simulated by a circuit with at most $\log(n+1)$ negations.
In 1974 Fischer has shown that this can be done with only
polynomial increase in size.

In this note we observe that some explicit monotone functions ... more >>>


TR04-062 | 28th July 2004
Stasys Jukna

A note on the P versus NP intersected with co-NP question in communication complexity

Revisions: 1 , Comments: 1

We consider the P versus NP\cap coNP question for the classical two-party communication protocols: if both a boolean function and its negation have small nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition case this question was answered by Aho, Ullman and ... more >>>


TR02-004 | 2nd November 2001
Till Tantau

A Note on the Power of Extra Queries to Membership Comparable Sets

A language is called k-membership comparable if there exists a
polynomial-time algorithm that excludes for any k words one of
the 2^k possibilities for their characteristic string.
It is known that all membership comparable languages can be
reduced to some P-selective language with polynomially many
adaptive queries. We show however ... more >>>


TR12-121 | 25th September 2012
Pavel Hrubes

A note on the real $\tau$-conjecture and the distribution of roots

Revisions: 2

Koiran's real $\tau$-conjecture asserts that if a non-zero real polynomial can be written as $f=\sum_{i=1}^{p}\prod_{j=1}^{q}f_{ij},$
where each $f_{ij}$ contains at most $k$ monomials, then the number of distinct real roots of $f$ is polynomial in $pqk$. We show that the conjecture implies quite a strong property of the ... more >>>


TR98-042 | 27th July 1998
Pavel Pudlak

A Note On the Use of Determinant for Proving Lower Bounds on the Size of Linear Circuits

Comments: 1


We consider computations of linear forms over {\bf R} by
circuits with linear gates where the absolute values
coefficients are bounded by a constant. Also we consider a
related concept of restricted rigidity of a matrix. We prove
some lower bounds on the size of such circuits and the
more >>>


TR16-032 | 10th March 2016
Roei Tell

A Note on Tolerant Testing with One-Sided Error

A tolerant tester with one-sided error for a property is a tester that accepts every input that is close to the property, with probability 1, and rejects every input that is far from the property, with positive probability. In this note we show that such testers require a linear number ... more >>>


TR04-118 | 21st December 2004
Marek Karpinski, Yakov Nekrich

A Note on Traversing Skew Merkle Trees

We consider the problem of traversing skew (unbalanced) Merkle
trees and design an algorithm for traversing a skew Merkle tree
in time O(log n/log t) and space O(log n (t/log t)), for any choice
of parameter t\geq 2.
This algorithm can be of special interest in situations when
more >>>


TR96-023 | 21st March 1996
Eric Allender

A Note on Uniform Circuit Lower Bounds for the Counting Hierarchy

Comments: 1

A very recent paper by Caussinus, McKenzie, Therien, and Vollmer
[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0
is properly contained in the counting hierarchy. Thus, [CMTV] shows
that there are problems in ModPH that require superpolynomial-size
uniform ACC^0 ... more >>>


TR07-016 | 13th February 2007
Prasad Raghavendra

A Note on Yekhanin's Locally Decodable Codes

Revisions: 1

Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors.

In a recent breakthrough, Yekhanin designed $3$-query LDCs that hugely improve over earlier constructions. Specifically, for a Mersenne prime $p ... more >>>


TR16-060 | 15th April 2016
Henry Yuen

A parallel repetition theorem for all entangled games

The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known.
... more >>>


TR09-027 | 2nd April 2009
Iftach Haitner

A Parallel Repetition Theorem for Any Interactive Argument

Revisions: 1

The question whether or not parallel repetition reduces the soundness error is a fundamental question in the theory of protocols. While parallel repetition reduces (at an exponential rate) the error in interactive proofs and (at a weak exponential rate) in special cases of interactive arguments (e.g., 3-message protocols - Bellare, ... more >>>


TR10-019 | 19th February 2010
Andrew Drucker

A PCP Characterization of AM

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class $\mathsf{AM}$. This gives a `PCP characterization' of $\mathsf{AM}$ analogous to the PCP Theorem for $\mathsf{NP}$. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, ... more >>>


TR14-084 | 12th June 2014
Luke Schaeffer

A Physically Universal Cellular Automaton

Several cellular automata (CA) are known to be universal in the sense that one can simulate arbitrary computations (e.g., circuits or Turing machines) by carefully encoding the computational device and its input into the cells of the CA. In this paper, we consider a different kind of universality proposed by ... more >>>


TR17-026 | 17th February 2017
Valentine Kabanets, Daniel Kane, Zhenjian Lu

A Polynomial Restriction Lemma with Applications

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... more >>>


TR00-064 | 29th August 2000
Klaus Jansen, Marek Karpinski, Andrzej Lingas

A Polynomial Time Approximation Scheme for MAX-BISECTION on Planar Graphs

The Max-Bisection and Min-Bisection are the problems of finding
partitions of the vertices of a given graph into two equal size subsets so as
to maximize or minimize, respectively, the number of edges with exactly one
endpoint in each subset.
In this paper we design the first ... more >>>


TR02-041 | 2nd July 2002
Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon

A Polynomial Time Approximation Scheme for Metric MIN-BISECTION

We design a polynomial time approximation scheme (PTAS) for
the problem of Metric MIN-BISECTION of dividing a given finite metric
space into two halves so as to minimize the sum of distances across
that partition. The method of solution depends on a new metric placement
partitioning ... more >>>


TR02-044 | 16th July 2002
Wenceslas Fernandez de la Vega, Marek Karpinski

A Polynomial Time Approximation Scheme for Subdense MAX-CUT

We prove that the subdense instances of MAX-CUT of average
degree Omega(n/logn) posses a polynomial time approximation scheme (PTAS).
We extend this result also to show that the instances of general 2-ary
maximum constraint satisfaction problems (MAX-CSP) of the same average
density have PTASs. Our results ... more >>>


TR10-088 | 17th May 2010
Jiri Sima, Stanislav Zak

A Polynomial Time Construction of a Hitting Set for Read-Once Branching Programs of Width 3

Revisions: 2 , Comments: 3

The relationship between deterministic and probabilistic computations is one of the central issues in complexity theory. This problem can be tackled by constructing polynomial time hitting set generators which, however, belongs to the hardest problems in computer science even for severely restricted computational models. In our work, we consider read-once ... more >>>


TR04-038 | 27th April 2004
John Case, Sanjay Jain, Rüdiger Reischuk, Frank Stephan, Thomas Zeugmann

A Polynomial Time Learner for a Subclass of Regular Patterns

Presented is an algorithm (for learning a subclass of erasing regular
pattern languages) which
can be made to run with arbitrarily high probability of
success on extended regular languages generated by patterns
$\pi$ of the form $x_0 \alpha_1 x_1 ... \alpha_m x_m$
for unknown $m$ but known $c$,
more >>>


TR00-079 | 12th September 2000
Mark Jerrum, Eric Vigoda

A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries

We present a fully-polynomial randomized approximation scheme
for computing the permanent of an arbitrary matrix
with non-negative entries.

more >>>

TR09-078 | 16th September 2009
Falk Unger

A Probabilistic Inequality with Applications to Threshold Direct-product Theorems

We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments.

This inequality allows us to simplify and strengthen several known ... more >>>


TR98-051 | 20th July 1998
Petr Savicky

A probabilistic nonequivalence test for syntactic (1,+k)-branching programs


Branching programs are a model for representing Boolean
functions. For general branching programs, the
satisfiability and nonequivalence tests are NP-complete.
For read-once branching programs, which can test each
variable at most once in each computation, the satisfiability
test is trivial and there is also a probabilistic polynomial
time test ... more >>>


TR05-103 | 17th August 2005
Leonid Gurvits

A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables .
Assume that this polynomial satisfies the property : \\

$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$ . \\

We prove that ... more >>>


TR04-063 | 23rd July 2004
Yehuda Lindell, Benny Pinkas

A Proof of Yao's Protocol for Secure Two-Party Computation

Revisions: 1

In the mid 1980's, Yao presented a constant-round protocol for
securely computing any two-party functionality in the presence of
semi-honest adversaries (FOCS 1986). In this paper, we provide a
complete description of Yao's protocol, along with a rigorous
proof of security. Despite the importance of Yao's protocol to the
field ... more >>>


TR96-065 | 13th December 1996
Miklos Ajtai, Cynthia Dwork

A Public-Key Cryptosystem with Worst-Case/Average-Case Equivalence

Revisions: 1 , Comments: 1

We present a probabilistic public key cryptosystem which is
secure unless the following worst-case lattice problem can be solved in
polynomial time:
"Find the shortest nonzero vector in an n dimensional lattice
L where the shortest vector v is unique in the sense that any other
vector whose ... more >>>


TR17-028 | 17th February 2017
Mrinal Kumar

A quadratic lower bound for homogeneous algebraic branching programs

Revisions: 1

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex $s$, and end vertex $t$ and each edge having a weight which is an affine form in $\F[x_1, x_2, \ldots, x_n]$. An ABP computes a polynomial in a natural way, as the sum of weights of ... more >>>


TR03-086 | 1st December 2003
Amos Beimel, Tal Malkin

A Quantitative Approach to Reductions in Secure Computation

Secure computation is one of the most fundamental cryptographic tasks.
It is known that all functions can be computed securely in the
information theoretic setting, given access to a black box for some
complete function such as AND. However, without such a black box, not
all functions can be securely ... more >>>


TR08-017 | 16th December 2007
Thomas Watson, Dieter van Melkebeek

A Quantum Time-Space Lower Bound for the Counting Hierarchy

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real $d$ and every positive real $\epsilon$ ... more >>>


TR09-074 | 10th September 2009
Suguru Tamaki, Yuichi Yoshida

A Query Efficient Non-Adaptive Long Code Test with Perfect Completeness

Long Code testing is a fundamental problem in the area of property
testing and hardness of approximation.
Long Code is a function of the form $f(x)=x_i$ for some index $i$.
In the Long Code testing, the problem is, given oracle access to a
collection of Boolean functions, to decide whether ... more >>>


TR05-156 | 13th December 2005
Jonathan A. Kelner, Daniel A. Spielman

A Randomized Polynomial-Time Simplex Algorithm for Linear Programming (Preliminary Version)

We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input.

We begin by reducing the input linear program to a special form in which we ... more >>>


TR05-107 | 28th September 2005
Avi Wigderson, David Xiao

A Randomness-Efficient Sampler for Matrix-valued Functions and Applications

Revisions: 1

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter, in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is ... more >>>


TR12-124 | 29th September 2012
Massimo Lauria

A rank lower bound for cutting planes proofs of Ramsey Theorem

Ramsey Theorem is a cornerstone of combinatorics and logic. In its
simplest formulation it says that there is a function $r$ such that
any simple graph with $r(k,s)$ vertices contains either a clique of
size $k$ or an independent set of size $s$. We study the complexity
of proving upper ... more >>>


TR05-035 | 24th March 2005
Christian Glaßer, Stephen Travers, Klaus W. Wagner

A Reducibility that Corresponds to Unbalanced Leaf-Language Classes

We introduce the polynomial-time tree reducibility
(ptt-reducibility). Our main result states that for
languages $B$ and $C$ it holds that
$B$ ptt-reduces to $C$ if and only if
the unbalanced leaf-language class of $B$ is robustly contained in
the unbalanced leaf-language class of $C$.
... more >>>


TR17-027 | 16th February 2017
Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, Amnon Ta-Shma

A reduction from efficient non-malleable extractors to low-error two-source extractors with arbitrary constant rate

We show a reduction from the existence of explicit t-non-malleable
extractors with a small seed length, to the construction of explicit
two-source extractors with small error for sources with arbitrarily
small constant rate. Previously, such a reduction was known either
when one source had entropy rate above half [Raz05] or ... more >>>


TR13-156 | 15th November 2013
Jan Krajicek

A reduction of proof complexity to computational complexity for $AC^0[p]$ Frege systems

Revisions: 2

We give a general reduction of lengths-of-proofs lower bounds for
constant depth Frege systems in DeMorgan language augmented by
a connective counting modulo a prime $p$
(the so called $AC^0[p]$ Frege systems)
to computational complexity
lower bounds for search tasks involving search trees branching upon
values of linear maps on ... more >>>


TR04-006 | 6th January 2004
Günter Hotz

A remark on nondecidabilities of the initial value problem of ODEs

We prove that it is not decidable on R-machines if for a fixed finite intervall [a,b) the solution of the initial value problems of systems of ordinary differetial equations have solutions over this interval. This result holds independly from assumptions about differentiability of the right sides of the ODEs. Futhermore ... more >>>


TR12-005 | 13th January 2012
Periklis Papakonstantinou, Guang Yang

A remark on one-wayness versus pseudorandomness

Every pseudorandom generator is in particular a one-way function. If we only consider part of the output of the
pseudorandom generator is this still one-way? Here is a general setting formalizing this question. Suppose
$G:\{0,1\}^n\rightarrow \{0,1\}^{\ell(n)}$ is a pseudorandom generator with stretch $\ell(n)> n$. Let $M_R\in\{0,1\}^{m(n)\times \ell(n)}$ be a linear ... more >>>


TR97-020 | 15th May 1997
Oded Goldreich

A Sample of Samplers -- A Computational Perspective on Sampling (survey).


We consider the problem of estimating the average of a huge set of values.
That is,
given oracle access to an arbitrary function $f:\{0,1\}^n\mapsto[0,1]$,
we need to estimate $2^{-n} \sum_{x\in\{0,1\}^n} f(x)$
upto an additive error of $\epsilon$.
We are allowed to employ a randomized algorithm which may ... more >>>


TR12-071 | 29th May 2012
Kazuhisa Seto, Suguru Tamaki

A Satisfiability Algorithm and Average-Case Hardness for Formulas over the Full Binary Basis

We present a moderately exponential time algorithm for the satisfiability of Boolean formulas over the full binary basis.
For formulas of size at most $cn$, our algorithm runs in time $2^{(1-\mu_c)n}$ for some constant $\mu_c>0$.
As a byproduct of the running time analysis of our algorithm,
we get strong ... more >>>


TR16-100 | 27th June 2016
Suguru Tamaki

A Satisfiability Algorithm for Depth Two Circuits with a Sub-Quadratic Number of Symmetric and Threshold Gates

We consider depth 2 unbounded fan-in circuits with symmetric and linear threshold gates. We present a deterministic algorithm that, given such a circuit with $n$ variables and $m$ gates, counts the number of satisfying assignments in time $2^{n-\Omega\left(\left(\frac{n}{\sqrt{m} \cdot \poly(\log n)}\right)^a\right)}$ for some constant $a>0$. Our algorithm runs in time ... more >>>


TR15-136 | 28th July 2015
Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, Junichi Teruyama

A Satisfiability Algorithm for Depth-2 Circuits with a Symmetric Gate at the Top and AND Gates at the Bottom

In this paper, we present a moderately exponential time algorithm for the circuit satisfiability problem of
depth-2 unbounded-fan-in circuits with an arbitrary symmetric gate at the top and AND gates at the bottom.
As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in ... more >>>


TR01-024 | 1st March 2001
Stephen Cook, Antonina Kolokolova

A second-order system for polynomial-time reasoning based on Graedel's theorem

We introduce a second-order system V_1-Horn of bounded arithmetic
formalizing polynomial-time reasoning, based on Graedel's
second-order Horn characterization of P. Our system has
comprehension over P predicates (defined by Graedel's second-order
Horn formulas), and only finitely many function symbols. Other
systems of polynomial-time reasoning either ... more >>>


TR16-149 | 23rd September 2016
Eli Ben-Sasson, iddo Ben-Tov, Ariel Gabizon, Michael Riabzev

A security analysis of Probabilistically Checkable Proofs

Comments: 1

Probabilistically Checkable Proofs (PCPs) [Babai et al. FOCS 90; Arora et al. JACM 98] can be used to construct asymptotically efficient cryptographic zero knowledge arguments of membership in any language in NEXP, with minimal communication complexity and computational effort on behalf of both prover and verifier [Babai et al. STOC ... more >>>


TR00-027 | 11th April 2000
Pavol Duris, Juraj Hromkovic, Katsushi Inoue

A Separation of Determinism, Las Vegas and Nondeterminism for Picture Recognition

The investigation of the computational power of randomized computations
is one of the central tasks of current complexity and algorithm theory.
In this paper for the first time a "strong" separation between the power
of determinism, Las Vegas randomization, and nondeterminism for a compu-
ting model is proved. The computing ... more >>>


TR10-060 | 5th April 2010
Dmitry Gavinsky, Alexander A. Sherstov

A Separation of NP and coNP in Multiparty Communication Complexity

We prove that NP$\ne$coNP and coNP$\nsubseteq$MA in the number-on-forehead model of multiparty communication complexity for up to $k=(1-\epsilon)\log n$ players, where $\epsilon>0$ is any constant. Specifically, we construct a function $F:(\zoon)^k\to\zoo$ with co-nondeterministic
complexity $O(\log n)$ and Merlin-Arthur
complexity $n^{\Omega(1)}$.
The problem was open for $k\geq3$.

more >>>

TR98-045 | 17th July 1998
Detlef Sieling

A Separation of Syntactic and Nonsyntactic (1,+k)-Branching Programs

For (1,+k)-branching programs and read-k-times branching
programs syntactic and nonsyntactic variants can be distinguished. The
nonsyntactic variants correspond in a natural way to sequential
computations with restrictions on reading the input while lower bound
proofs are easier or only known for the syntactic variants. In this
paper it is shown ... more >>>


TR13-017 | 23rd January 2013
Pratik Worah

A Short Excursion into Semi-Algebraic Hierarchies

This brief survey gives a (roughly) self-contained overview of some complexity theoretic results about semi-algebraic proof systems and related hierarchies and the strong connections between them. The article is not intended to be a detailed survey on "Lift and Project" type optimization hierarchies (cf. Chlamtac and Tulsiani) or related proof ... more >>>


TR13-176 | 8th December 2013
Daniel Kane, Osamu Watanabe

A Short Implicant of CNFs with Relatively Many Satisfying Assignments

Revisions: 1 , Comments: 1

Consider any Boolean function $F(X_1,\ldots,X_N)$ that has more than $2^{-N^{d}}$ satisfying assignments and that can be expressed by a CNF formula with at most $N^{1+e}$ clauses for some $d>0$ and $e>0$ such that $d+e$ is less than $1$ (*). Then how many variables do we need to fix in order ... more >>>


TR13-012 | 16th January 2013
Hasan Abasi, Nader Bshouty

A Simple Algorithm for Undirected Hamiltonicity

We develop a new algebraic technique that gives a simple randomized algorithm for the simple $k$-path problem with the same complexity $O^*(1.657^k)$ as in [A. Bj\"orklund. Determinant Sums for Undirected Hamiltonicity. FOCS 2010, pp. 173--182, (2010). A. Bj\"orklund, T. Husfeldt, P. Kaski, M. Koivisto. Narrow sieves for parameterized paths and ... more >>>


TR06-121 | 14th September 2006
Charanjit Jutla

A Simple Biased Distribution for Dinur's Construction


TR08-093 | 1st October 2008
Cristopher Moore, Alexander Russell

A simple constant-probability RP reduction from NP to Parity P

The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in $\P^\numP$ relies on the fact that, under mild technical conditions on the complexity class $\mathcal{C}$, we have $\exists \,\mathcal{C} \subset \BP \cdot \oplus \,\mathcal{C}$. More concretely, there is a randomized reduction which transforms nonempty sets and the ... more >>>


TR00-030 | 31st May 2000

A Simple Model for Neural Computation with Firing Rates and Firing Correlations

A simple extension of standard neural network models is introduced that
provides a model for neural computations that involve both firing rates and
firing correlations. Such extension appears to be useful since it has been
shown that firing correlations play a significant computational role in
many biological neural systems. Standard ... more >>>


TR08-081 | 11th September 2008
Alexander Razborov

A simple proof of Bazzi's theorem

In 1990, Linial and Nisan asked if any polylog-wise independent distribution fools any function in AC^0. In a recent remarkable development, Bazzi solved this problem for the case of DNF formulas.
The aim of this note is to present a simplified version of his proof.

more >>>

TR15-080 | 7th May 2015
Noam Ta-Shma

A simple proof of the Isolation Lemma

We give a new simple proof for the Isolation Lemma, with slightly better parameters, that also gives non-trivial results even when the weight domain $m$ is smaller than the number of variables $n$.

more >>>

TR14-075 | 16th May 2014
Holger Dell

A simple proof that AND-compression of NP-complete problems is hard

Revisions: 3

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a simple proof of this result.

An AND-compression is ... more >>>


TR07-114 | 28th September 2007
Jakob Nordström

A Simplified Way of Proving Trade-off Results for Resolution

We present a greatly simplified proof of the length-space
trade-off result for resolution in Hertel and Pitassi (2007), and
also prove a couple of other theorems in the same vein. We point
out two important ingredients needed for our proofs to work, and
discuss possible conclusions to be drawn regarding ... more >>>


TR12-053 | 30th April 2012
Ankur Moitra

A Singly-Exponential Time Algorithm for Computing Nonnegative Rank

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in $r$. This algorithm (and earlier algorithms) are built on ... more >>>


TR09-047 | 20th April 2009
Eli Ben-Sasson, Jakob Nordström

A Space Hierarchy for k-DNF Resolution

Comments: 1

The k-DNF resolution proof systems are a family of systems indexed by
the integer k, where the kth member is restricted to operating with
formulas in disjunctive normal form with all terms of bounded arity k
(k-DNF formulas). This family was introduced in [Krajicek 2001] as an
extension of the ... more >>>


TR10-150 | 19th September 2010
Bjørn Kjos-Hanssen

A strong law of computationally weak subsets

We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if $X\in\mathcal A$ then $X$ has an infinite ... more >>>


TR10-141 | 18th September 2010 (removed)
Ran Raz

A Strong Parallel Repetition Theorem for Projection Games on Expanders


Reason: This paper has been remove on the author's behalf. Please note that TR10-142 is the corrected version.

TR10-142 | 18th September 2010
Ran Raz, Ricky Rosen

A Strong Parallel Repetition Theorem for Projection Games on Expanders

The parallel repetition theorem states that for any Two
Prover Game with value at most $1-\epsilon$ (for $\epsilon<1/2$),
the value of the game repeated $n$ times in parallel is at most
$(1-\epsilon^3)^{\Omega(n/s)}$, where $s$ is the length of the
answers of the two provers. For Projection
Games, the bound on ... more >>>


TR13-133 | 23rd September 2013
Cassio P. de Campos, Georgios Stamoulis, Dennis Weyland

A Structured View on Weighted Counting with Relations to Quantum Computation and Applications

Revisions: 1

Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path and the goal is to compute the sum of the weights of all paths (instead of computing the number of accepting paths). We present a structured view on weighted counting by ... more >>>


TR95-060 | 21st November 1995
Nader H. Bshouty

A Subexponential Exact Learning Algorithm for DNF Using Equivalence Queries


We present a $2^{\tilde O(\sqrt{n})}$ time exact learning
algorithm for polynomial size
DNF using equivalence queries only. In particular, DNF
is PAC-learnable in subexponential time under any distribution.
This is the first subexponential time
PAC-learning algorithm for DNF under any distribution.

more >>>

TR97-050 | 27th October 1997
Stanislav Zak

A subexponential lower bound for branching programs restricted with regard to some semantic aspects

Branching programs (b.p.s) or binary decision diagrams are a
general graph-based model of sequential computation. The b.p.s of
polynomial size are a nonuniform counterpart of LOG. Lower bounds
for different kinds of restricted b.p.s are intensively
investigated. The restrictions based on the number of tests of
more >>>


TR15-108 | 30th June 2015
Shalev Ben-David

A Super-Grover Separation Between Randomized and Quantum Query Complexities

We construct a total Boolean function $f$ satisfying
$R(f)=\tilde{\Omega}(Q(f)^{5/2})$, refuting the long-standing
conjecture that $R(f)=O(Q(f)^2)$ for all total Boolean functions.
Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions,
we improve this to $R(f)=\tilde{\Omega}(Q(f)^3)$.
Our construction is motivated by the Göös-Pitassi-Watson function
but does not ... more >>>


TR13-091 | 17th June 2013
Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

A super-polynomial lower bound for regular arithmetic formulas.

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>


TR17-001 | 6th January 2017
Stephen Cook, Bruce Kapron

A Survey of Classes of Primitive Recursive Functions

This paper is a transcription of mimeographed course notes titled ``A Survey of Classes of Primitive Recursive Functions", by S.A. Cook, for the University of California Berkeley course Math 290, Sect. 14, January 1967. The notes present a survey of subrecursive function
classes (and classes of relations based on these ... more >>>


TR07-099 | 30th September 2007
Dieter van Melkebeek

A Survey of Lower Bounds for Satisfiability and Related Problems

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by ... more >>>


TR15-063 | 15th April 2015
Clement Canonne

A Survey on Distribution Testing: Your Data is Big. But is it Blue?

Revisions: 1

The field of property testing originated in work on program checking, and has evolved into an established and very active research area. In this work, we survey the developments of one of its most recent and prolific offspring, distribution testing. This subfield, at the junction of property testing and Statistics, ... more >>>


TR12-051 | 25th April 2012
Dmitry Gavinsky, Shachar Lovett, Michael Saks, Srikanth Srinivasan

A Tail Bound for Read-k Families of Functions

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,\ldots,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,\ldots,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,\ldots,Y_r$.

more >>>

TR10-145 | 21st September 2010
Ron Rothblum

A Taxonomy of Enhanced Trapdoor Permutations

Trapdoor permutations (TDPs) are among the most widely studied
building blocks of cryptography. Despite the extensive body of
work that has been dedicated to their study, in many setting and
applications (enhanced) trapdoor permutations behave
unexpectedly. In particular, a TDP may become easy to invert when
the inverter is given ... more >>>


TR09-075 | 17th September 2009
Oded Goldreich, Brendan Juba, Madhu Sudan

A Theory of Goal-Oriented Communication

Revisions: 1 , Comments: 1

We put forward a general theory of goal-oriented communication, where communication is not an end in itself, but rather a means to achieving some goals of the communicating parties. The goals can vary from setting to setting, and we provide a general framework for describing any such goal. In this ... more >>>


TR98-034 | 23rd June 1998
Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan

A tight characterization of NP with 3 query PCPs


It is known that there exists a PCP characterization of NP
where the verifier makes 3 queries and has a {\em one-sided}
error that is bounded away from 1; and also that 2 queries
do not suffice for such a characterization. Thus PCPs with
3 ... more >>>


TR14-027 | 21st February 2014
Andris Ambainis, Krisjanis Prusis

A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity

Revisions: 1

Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity ... more >>>


TR11-086 | 2nd June 2011
Masaki Yamamoto

A tighter lower bound on the circuit size of the hardest Boolean functions

In [IPL2005],
Frandsen and Miltersen improved bounds on the circuit size $L(n)$ of the hardest Boolean function on $n$ input bits:
for some constant $c>0$:
\[
\left(1+\frac{\log n}{n}-\frac{c}{n}\right)
\frac{2^n}{n}
\leq
L(n)
\leq
\left(1+3\frac{\log n}{n}+\frac{c}{n}\right)
\frac{2^n}{n}.
\]
In this note,
we announce a modest ... more >>>


TR17-020 | 12th February 2017
Ran Raz

A Time-Space Lower Bound for a Large Class of Learning Problems

We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples.

Our result is stated in terms of the norm ... more >>>


TR04-073 | 9th July 2004
Henning Fernau

A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set

In this paper, we show how to systematically
improve on parameterized algorithms and their
analysis, focusing on search-tree based algorithms
for d-Hitting Set, especially for d=3.
We concentrate on algorithms which are easy to implement,
in contrast with the highly sophisticated algorithms
which have been elsewhere designed to ... more >>>


TR14-137 | 24th October 2014
Neil Thapen

A trade-off between length and width in resolution

We describe a family of CNF formulas in $n$ variables, with small initial width, which have polynomial length resolution refutations. By a result of Ben-Sasson and Wigderson it follows that they must also have narrow resolution refutations, of width $O(\sqrt {n \log n})$. We show that, for our formulas, this ... more >>>


TR03-075 | 7th September 2003
Agostino Capponi

A tutorial on the Deterministic two-party Communication Complexity

Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of $f$. This complexity measure was introduced by Yao \cite{1} ... more >>>


TR01-016 | 22nd December 2000
Ran Canetti

A unified framework for analyzing security of protocols

Revisions: 5

Building on known definitions, we present a unified general framework for
defining and analyzing security of cryptographic protocols. The framework
allows specifying the security requirements of a large number of
cryptographic tasks, such as signature, encryption, authentication, key
exchange, commitment, oblivious transfer, zero-knowledge, secret sharing,
general function evaluation, and ... more >>>


TR10-161 | 25th October 2010
Arnab Bhattacharyya, Elena Grigorescu, Asaf Shapira

A Unified Framework for Testing Linear-Invariant Properties

The study of the interplay between the testability of properties of Boolean functions and the invariances acting on their domain which preserve the property was initiated by Kaufman and Sudan (STOC 2008). Invariance with respect to F_2-linear transformations is arguably the most common symmetry exhibited by natural properties of Boolean ... more >>>


TR16-186 | 19th November 2016
Jayadev Acharya, Hirakendu Das, Alon Orlitsky, Ananda Theertha Suresh

A Unified Maximum Likelihood Approach for Optimal Distribution Property Estimation

The advent of data science has spurred interest in estimating properties of discrete distributions over large alphabets. Fundamental symmetric properties such as support size, support coverage, entropy, and proximity to uniformity, received most attention, with each property estimated using a different technique and often intricate analysis tools.

Motivated by the ... more >>>


TR17-019 | 8th February 2017
Andreas Krebs, Nutan Limaye, Michael Ludwig

A Unified Method for Placing Problems in Polylogarithmic Depth

Revisions: 1

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem ... more >>>


TR13-101 | 12th July 2013
Colin Jia Zheng, Salil Vadhan

A Uniform Min-Max Theorem with Applications in Cryptography

Revisions: 2

We present a new, more constructive proof of von Neumann's Min-Max Theorem for two-player zero-sum game --- specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of ... more >>>


TR14-103 | 8th August 2014
Uriel Feige, Michal Feldman, Nicole Immorlica, Rani Izsak, Lucier Brendan, Vasilis Syrgkanis

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as $MPH$ (Maximum over Positive Hypergraphs).
Levels of the hierarchy correspond to the degree of complementarity in a given function.
The highest level of the hierarchy, $MPH$-$m$ (where $m$ is the total number of items) captures all ... more >>>


TR05-078 | 25th May 2005
Kooshiar Azimian, Javad Mohajeri, Mahmoud Salmasizadeh, Siamak Fayyaz

A Verifiable Partial Key Escrow, Based on McCurley Encryption Scheme

Revisions: 1

In this paper, firstly we propose two new concepts concerning the notion of key escrow encryption schemes: provable partiality and independency. Roughly speaking we say that a scheme has provable partiality if existing polynomial time algorithm for recovering the secret knowing escrowed information implies a polynomial time algorithm that can ... more >>>


TR02-033 | 11th June 2002
Beate Bollig

A very simple function that requires exponential size nondeterministic graph-driven read-once branching programs

Branching programs are a well-established computation
model for boolean functions, especially read-once
branching programs (BP1s) have been studied intensively.
A very simple function $f$ in $n^2$ variables is
exhibited such that both the function $f$ and its negation
$\neg f$ can be computed by $\Sigma^3_p$-circuits,
the ... more >>>


TR13-029 | 19th February 2013
Deeparnab Chakrabarty, C. Seshadhri

A {\huge ${o(n)}$} monotonicity tester for Boolean functions over the hypercube

Revisions: 1

Given oracle access to a Boolean function $f:\{0,1\}^n \mapsto \{0,1\}$, we design a randomized tester that takes as input a parameter $\eps>0$, and outputs {\sf Yes} if the function is monotone, and outputs {\sf No} with probability $>2/3$, if the function is $\eps$-far from monotone. That is, $f$ needs to ... more >>>


TR13-030 | 20th February 2013
Elad Haramaty, Noga Ron-Zewi, Madhu Sudan

Absolutely Sound Testing of Lifted Codes

In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of "affine-invariant'' codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes ... more >>>


TR11-050 | 11th April 2011
Claus-Peter Schnorr

Accelerated Slide- and LLL-Reduction

Revisions: 7

Given an LLL-basis $B$ of dimension $n= hk$ we accelerate slide-reduction with blocksize $k$ to run under a reasonable assjmption in \
$\frac1{6} \, n^2 h \,\log_{1+\varepsilon} \, \alpha $ \
local SVP-computations in dimension $k$, where $\alpha \ge \frac 43$
measures the quality of the ... more >>>


TR07-088 | 7th September 2007
Elad Hazan, C. Seshadhri

Adaptive Algorithms for Online Decision Problems

Revisions: 1

We study the notion of learning in an oblivious changing environment. Existing online learning algorithms which minimize regret are shown to converge to the average of all locally optimal solutions. We propose a new performance metric, strengthening the standard metric of regret, to capture convergence to locally optimal solutions, and ... more >>>


TR06-042 | 16th March 2006
Amit Deshpande, Santosh Vempala

Adaptive Sampling and Fast Low-Rank Matrix Approximation

We prove that any real matrix $A$ contains a subset of at most
$4k/\eps + 2k \log(k+1)$ rows whose span ``contains" a matrix of
rank at most $k$ with error only $(1+\eps)$ times the error of the
best rank-$k$ approximation of $A$. This leads to an algorithm to
find such ... more >>>


TR96-051 | 1st October 1996
Richard Beigel, William Gasarch, Ming Li, Louxin Zhang

Addition in $\log_2{n}$ + O(1)$ Steps on Average: A Simple Analysis

We demonstrate the use of Kolmogorov complexity in average case
analysis of algorithms through a classical example: adding two $n$-bit
numbers in $\ceiling{\log_2{n}}+2$ steps on average. We simplify the
analysis of Burks, Goldstine, and von Neumann in 1946 and
(in more complete forms) of Briley and of Schay.

more >>>

TR15-123 | 31st July 2015
Xi Chen, Igor Carboni Oliveira, Rocco Servedio

Addition is exponentially harder than counting for shallow monotone circuits

Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes ... more >>>


TR14-044 | 2nd April 2014
Daniel Dewey

Additively efficient universal computers

We give evidence for a stronger version of the extended Church-Turing thesis: among the set of physically possible computers, there are computers that can simulate any other realizable computer with only additive constant overhead in space, time, and other natural resources. Complexity-theoretic results that hold for these computers can therefore ... more >>>


TR11-008 | 27th January 2011
Scott Aaronson, Andrew Drucker

Advice Coins for Classical and Quantum Computation

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states ... more >>>


TR11-120 | 6th September 2011
Thomas Watson

Advice Lower Bounds for the Dense Model Theorem

Revisions: 1

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution $D$ that is $\delta$-dense in a distribution that is ... more >>>


TR10-044 | 12th March 2010
Eli Ben-Sasson, Swastik Kopparty

Affine Dispersers from Subspace Polynomials

{\em Dispersers} and {\em extractors} for affine sources of dimension $d$ in $\mathbb F_p^n$ --- where $\mathbb F_p$ denotes the finite field of prime size $p$ --- are functions $f: \mathbb F_p^n \rightarrow \mathbb F_p$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq ... more >>>


TR14-010 | 23rd January 2014
Jean Bourgain, Zeev Dvir, Ethan Leeman

Affine extractors over large fields with exponential error

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.

more >>>

TR11-061 | 18th April 2011
Neeraj Kayal

Affine projections of polynomials

Revisions: 1

An $m$-variate polynomial $f$ is said to be an affine projection of some $n$-variate polynomial $g$ if there exists an $n \times m$ matrix $A$ and an $n$-dimensional vector $b$ such that $f(x) = g(A x + b)$. In other words, if $f$ can be obtained by replacing each variable ... more >>>


TR01-035 | 15th April 2001
Amir Shpilka

Affine Projections of Symmetric Polynomials

In this paper we introduce a new model for computing=20
polynomials - a depth 2 circuit with a symmetric gate at the top=20
and plus gates at the bottom, i.e the circuit computes a=20
symmetric function in linear functions -
$S_{m}^{d}(\ell_1,\ell_2,...,\ell_m)$ ($S_{m}^{d}$ is the $d$'th=20
elementary symmetric polynomial in $m$ ... more >>>


TR16-040 | 16th March 2016
Baris Aydinlioglu, Eric Bach

Affine Relativization: Unifying the Algebrization and Relativization Barriers

Revisions: 1

We strengthen existing evidence for the so-called "algebrization barrier". Algebrization --- short for algebraic relativization --- was introduced by Aaronson and Wigderson (AW) in order to characterize proofs involving arithmetization, simulation, and other "current techniques". However, unlike relativization, eligible statements under this notion do not seem to have basic closure ... more >>>


TR15-179 | 10th November 2015
Divesh Aggarwal, Kaave Hosseini, Shachar Lovett

Affine-malleable Extractors, Spectrum Doubling, and Application to Privacy Amplification

The study of seeded randomness extractors is a major line of research in theoretical computer science. The goal is to construct deterministic algorithms which can take a ``weak" random source $X$ with min-entropy $k$ and a uniformly random seed $Y$ of length $d$, and outputs a string of length close ... more >>>


TR15-009 | 16th January 2015
Aloni Cohen, Shafi Goldwasser, Vinod Vaikuntanathan

Aggregate Pseudorandom Functions and Connections to Learning

Revisions: 1

In the first part of this work, we introduce a new type of pseudo-random function for which ``aggregate queries'' over exponential-sized sets can be efficiently answered. An example of an aggregate query may be the product of all function values belonging to an exponential-sized interval, or the sum of all ... more >>>


TR04-028 | 19th March 2004
Arfst Nickelsen, Till Tantau, Lorenz Weizsäcker

Aggregates with Component Size One Characterize Polynomial Space

Aggregates are a computational model similar to circuits, but the
underlying graph is not necessarily acyclic. Logspace-uniform
polynomial-size aggregates decide exactly the languages in PSPACE;
without uniformity condition they decide the languages in
PSPACE/poly. As a measure of similarity to boolean circuits we
introduce the parameter component size. We ... more >>>


TR10-185 | 2nd December 2010
Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu

Agnostic Learning of Monomials by Halfspaces is Hard

We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples, it is $\mathrm{NP}$-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples, for arbitrary constants $\epsilon ... more >>>


TR94-020 | 12th December 1994

Agnostic PAC-Learning of Functions on Analog Neural Nets

We consider learning on multi-layer neural nets with piecewise polynomial
activation functions and a fixed number k of numerical inputs. We exhibit
arbitrarily large network architectures for which efficient and provably
successful learning algorithms exist in the rather realistic refinement of
Valiant's model for probably approximately correct learning ("PAC-learning")
where ... more >>>


TR00-013 | 14th February 2000
Daniel Král

Algebraic and Uniqueness Properties of Parity Ordered Binary Decision Diagrams and their Generalization

Ordered binary decision diagrams (OBDDs) and parity ordered binary
decision diagrams have proved their importance as data structures
representing Boolean functions. In addition to parity OBDDs we study
their generalization which we call parity AOBDDs, give the algebraic
characterization theorem and compare their minimal size to the size
more >>>


TR15-172 | 3rd November 2015
Benny Applebaum, Shachar Lovett

Algebraic Attacks against Random Local Functions and Their Countermeasures

Revisions: 1

Suppose that you have $n$ truly random bits $x=(x_1,\ldots,x_n)$ and you wish to use them to generate $m\gg n$ pseudorandom bits $y=(y_1,\ldots, y_m)$ using a local mapping, i.e., each $y_i$ should depend on at most $d=O(1)$ bits of $x$. In the polynomial regime of $m=n^s$, $s>1$, the only known solution, ... more >>>


TR14-130 | 17th October 2014
Ankit Gupta

Algebraic Geometric Techniques for Depth-4 PIT & Sylvester-Gallai Conjectures for Varieties

Revisions: 1

We present an algebraic-geometric approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-4 circuits with bounded top fanin. Using our approach, we devise such an algorithm for the case when such circuits have bounded bottom fanin and satisfy a certain non-degeneracy condition. In particular, we ... more >>>


TR11-022 | 14th February 2011
Malte Beecken, Johannes Mittmann, Nitin Saxena

Algebraic Independence and Blackbox Identity Testing

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials $\{f_1,\ldots, f_m\} \subset \mathbb{F}[x_1,\ldots, x_n]$ are called algebraically independent if there is no non-zero polynomial $F$ such that $F(f_1, \ldots, f_m) = 0$. The transcendence degree, $\mbox{trdeg}\{f_1,\ldots, f_m\}$, is the maximal ... more >>>


TR12-014 | 20th February 2012
Johannes Mittmann, Nitin Saxena, Peter Scheiblechner

Algebraic Independence in Positive Characteristic -- A p-Adic Calculus

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic $p>0$, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to ... more >>>


TR07-022 | 20th February 2007
Rafail Ostrovsky, William Skeith

Algebraic Lower Bounds for Computing on Encrypted Data

In cryptography, there has been tremendous success in building
primitives out of homomorphic semantically-secure encryption
schemes, using homomorphic properties in a black-box way. A few
notable examples of such primitives include items like private
information retrieval schemes and collision-resistant hash functions. In this paper, we illustrate a general
methodology for ... more >>>


TR16-101 | 1st July 2016
Toniann Pitassi, Iddo Tzameret

Algebraic Proof Complexity: Progress, Frontiers and Challenges

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight ... more >>>


TR01-011 | 21st January 2001
Dima Grigoriev, Edward Hirsch

Algebraic proof systems over formulas

We introduce two algebraic propositional proof systems F-NS
and F-PC. The main difference of our systems from (customary)
Nullstellensatz and Polynomial Calculus is that the polynomials
are represented as arbitrary formulas (rather than sums of
monomials). Short proofs of Tseitin's tautologies in the
constant-depth version of F-NS provide ... more >>>


TR10-097 | 16th June 2010
Iddo Tzameret

Algebraic Proofs over Noncommutative Formulas

Revisions: 1

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege--yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in Buss ... more >>>


TR07-111 | 1st November 2007
Tali Kaufman, Madhu Sudan

Algebraic Property Testing: The Role of Invariance

We argue that the symmetries of a property being tested play a
central role in property testing. We support this assertion in the
context of algebraic functions, by examining properties of functions
mapping a vector space $\K^n$ over a field $\K$ to a subfield $\F$.
We consider $\F$-linear properties that ... more >>>


TR05-132 | 8th November 2005
Venkatesan Guruswami

Algebraic-geometric generalizations of the Parvaresh-Vardy codes

This paper is concerned with a new family of error-correcting codes
based on algebraic curves over finite fields, and list decoding
algorithms for them. The basic goal in the subject of list decoding is
to construct error-correcting codes $C$ over some alphabet $\Sigma$
which have good rate $R$, and at ... more >>>


TR08-005 | 15th January 2008
Scott Aaronson, Avi Wigderson

Algebrization: A New Barrier in Complexity Theory

Any proof of P!=NP will have to overcome two barriers: relativization
and natural proofs. Yet over the last decade, we have seen circuit
lower bounds (for example, that PP does not have linear-size circuits)
that overcome both barriers simultaneously. So the question arises of
whether there ... more >>>


TR08-039 | 7th April 2008
Oded Goldreich, Dana Ron

Algorithmic Aspects of Property Testing in the Dense Graphs Model

In this paper we consider two refined questions regarding
the query complexity of testing graph properties
in the adjacency matrix model.
The first question refers to the relation between adaptive
and non-adaptive testers, whereas the second question refers to
testability within complexity that
is inversely proportional to ... more >>>


TR11-128 | 21st September 2011
Michael Elberfeld, Andreas Jakoby, Till Tantau

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

An algorithmic meta theorem for a logic and a class $C$ of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded ... more >>>


TR09-147 | 30th December 2009
Stephan Kreutzer

Algorithmic Meta-Theorems

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a logical and a
structural component, that is they are results of the form:
"every computational problem that can be formalised in a given logic L ... more >>>


TR10-073 | 21st April 2010
Neeraj Kayal

Algorithms for Arithmetic Circuits

Given a multivariate polynomial f(x) in F[x] as an arithmetic circuit we would like to efficiently determine:

(i) [Identity Testing.] Is f(x) identically zero?

(ii) [Degree Computation.] Is the degree of the
polynomial f(x) at most a given integer d.

(iii) [Polynomial Equivalence.] Upto an ... more >>>


TR05-033 | 5th March 2005
Martin Furer, Shiva Prasad Kasiviswanathan

Algorithms for Counting 2-SAT Solutions and Colorings with Applications

An algorithm is presented for counting the number of maximum weight satisfying assignments of a 2SAT formula. The worst case running time of $O(\mbox{poly($n$)} \cdot 1.2461^n)$ for formulas with $n$ variables improves on the previous bound of $O(\mbox{poly($n$)} \cdot 1.2561^n)$ by Dahll\"of, Jonsson, and Wahlstr\"om . The weighted 2SAT counting ... more >>>


TR13-123 | 6th September 2013
Joshua Grochow, Youming Qiao

Algorithms for group isomorphism via group extensions and cohomology

The isomorphism problem for groups given by multiplication tables (GpI) is well-known to be solvable in n^O(log n) time, but only recently has there been significant progress towards polynomial time. For example, in 2012 Babai et al. (ICALP 2012) gave a polynomial-time algorithm for groups with no abelian normal subgroups. ... more >>>


TR07-059 | 6th July 2007
Shankar Kalyanaraman, Chris Umans

Algorithms for Playing Games with Limited Randomness

We study multiplayer games in which the participants have access to
only limited randomness. This constrains both the algorithms used to
compute equilibria (they should use little or no randomness) as well
as the mixed strategies that the participants are capable of playing
(these should be sparse). We frame algorithmic ... more >>>


TR01-012 | 4th January 2001
Evgeny Dantsin, Edward Hirsch, Sergei Ivanov, Maxim Vsemirnov

Algorithms for SAT and Upper Bounds on Their Complexity

We survey recent algorithms for the propositional
satisfiability problem, in particular algorithms
that have the best current worst-case upper bounds
on their complexity. We also discuss some related
issues: the derandomization of the algorithm of
Paturi, Pudlak, Saks and Zane, the Valiant-Vazirani
Lemma, and random walk ... more >>>


TR03-072 | 15th September 2003
Evgeny Dantsin, Edward Hirsch, Alexander Wolpert

Algorithms for SAT based on search in Hamming balls

We present a simple randomized algorithm for SAT and prove an upper
bound on its running time. Given a Boolean formula F in conjunctive
normal form, the algorithm finds a satisfying assignment for F
(if any) by repeating the following: Choose an assignment A at
random and ... more >>>


TR10-122 | 18th July 2010
Zhixiang Chen, Bin Fu, Yang Liu, Robert Schweller

Algorithms for Testing Monomials in Multivariate Polynomials

This paper is our second step towards developing a theory of
testing monomials in multivariate polynomials. The central
question is to ask whether a polynomial represented by an
arithmetic circuit has some types of monomials in its sum-product
expansion. The complexity aspects of this problem and its variants
have been ... more >>>


TR11-124 | 15th September 2011
Nader Bshouty, Hanna Mazzawi

Algorithms for the Coin Weighing Problems with the Presence of Noise

The coin weighing problem is the following: Given $n$ coins for which $m$ of them are counterfeit with the same weight. The problem is to detect the counterfeit coins with minimal number of weighings. This problem has many applications in compressed sensing, multiple access adder channels, etc. The problem was ... more >>>


TR16-008 | 26th January 2016
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova

Algorithms from Natural Lower Bounds

Circuit analysis algorithms such as learning, SAT, minimum circuit size, and compression imply circuit lower bounds. We show a generic implication in the opposite direction: natural properties (in the sense of Razborov and Rudich) imply randomized learning and compression algorithms. This is the first such implication outside of the derandomization ... more >>>


TR13-117 | 1st September 2013
Igor Oliveira

Algorithms versus Circuit Lower Bounds

Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability ... more >>>


TR16-168 | 2nd November 2016
Eric Blais, Clement Canonne, Tom Gur

Alice and Bob Show Distribution Testing Lower Bounds (They don't talk to each other anymore.)

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method ... more >>>


TR06-122 | 20th September 2006
Noam Livne

All Natural NPC Problems Have Average-Case Complete Versions

Revisions: 1

In 1984 Levin put forward a suggestion for a theory of {\em average
case complexity}. In this theory a problem, called a {\em
distributional problem}, is defined as a pair consisting of a
decision problem and a probability distribution over the instances.
Introducing adequate notions of simple distributions and average
more >>>


TR97-040 | 17th September 1997
Dorit Dor, Shay Halperin, Uri Zwick

All Pairs Almost Shortest Paths

Let G=(V,E) be an unweighted undirected graph on n vertices. A simple
argument shows that computing all distances in G with an additive
one-sided error of at most 1 is as hard as Boolean matrix
multiplication. Building on recent work of Aingworth, Chekuri and
Motwani, we describe an \tilde{O}(min{n^{3/2}m^{1/2},n^{7/3}) time
more >>>


TR00-060 | 17th August 2000
Uri Zwick

All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication

We present two new algorithms for solving the {\em All
Pairs Shortest Paths\/} (APSP) problem for weighted directed
graphs. Both algorithms use fast matrix multiplication algorithms.

The first algorithm
solves the APSP problem for weighted directed graphs in which the edge
weights are integers of small absolute value in ... more >>>


TR99-004 | 3rd February 1999
Valentine Kabanets

Almost $k$-Wise Independence and Boolean Functions Hard for Read-Once Branching Programs

Revisions: 1

Andreev et al.~\cite{ABCR97} give constructions of Boolean
functions (computable by polynomial-size circuits) that require large
read-once branching program (1-b.p.'s): a function in P that requires
1-b.p. of size at least $2^{n-\polylog(n)}$, a function in quasipolynomial
time that requires 1-b.p. of size at least $2^{n-O(\log n)}$, and a
function in LINSPACE ... more >>>


TR02-048 | 31st July 2002
Noga Alon, Oded Goldreich, Yishay Mansour

Almost $k$-wise independence versus $k$-wise independence


We say that a distribution over $\{0,1\}^n$
is almost $k$-wise independent
if its restriction to every $k$ coordinates results in a
distribution that is close to the uniform distribution.
A natural question regarding almost $k$-wise independent
distributions is how close they are to some $k$-wise
independent distribution. We show ... more >>>


TR05-010 | 8th December 2004
Olivier Powell

Almost Completeness in Small Complexity Classes

We constructively prove the existence of almost complete problems under logspace manyone reduction for some small complexity classes by exhibiting a parametrizable construction which yields, when appropriately setting the parameters, an almost complete problem for PSPACE, the class of space efficiently decidable problems, and for SUBEXP, the class of problems ... more >>>


TR16-187 | 21st November 2016
morris yau

Almost Cubic Bound for Depth Three Circuits in VP

Revisions: 3

In "An Almost Cubic Lower Bound for $\sum\prod\sum$ circuits in VP", [BLS16] present an infinite family of polynomials, $\{P_n\}_{n \in \mathbb{Z}^+}$, with $P_n$
on $N = \Theta(n polylog(n))$
variables with degree $N$ being in VP such that every
$\sum\prod\sum$ circuit computing $P_n$ is of size $\Omega\big(\frac{N^3}{2^{\sqrt{\log N}}}\big)$.
We ... more >>>


TR07-012 | 22nd January 2007
Shachar Lovett, Sasha Sodin

Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits

Revisions: 1

It is well known that $\R^N$ has subspaces of dimension
proportional to $N$ on which the $\ell_1$ norm is equivalent to the
$\ell_2$ norm; however, no explicit constructions are known.
Extending earlier work by Artstein--Avidan and Milman, we prove that
such a subspace can be generated using $O(N)$ random bits.

... more >>>

TR07-086 | 7th September 2007
Venkatesan Guruswami, James R. Lee, Alexander Razborov

Almost Euclidean subspaces of $\ell_1^N$ via expander codes

We give an explicit (in particular, deterministic polynomial time)
construction of subspaces $X
\subseteq \R^N$ of dimension $(1-o(1))N$ such that for every $x \in X$,
$$(\log N)^{-O(\log\log\log N)} \sqrt{N}\, \|x\|_2 \leq \|x\|_1 \leq \sqrt{N}\, \|x\|_2.$$
If we are allowed to use $N^{1/\log\log N}\leq N^{o(1)}$ random bits
and ... more >>>


TR11-049 | 9th April 2011
Noga Alon, Shachar Lovett

Almost k-wise vs. k-wise independent permutations, and uniformity for general group actions

A family of permutations in $S_n$ is $k$-wise independent if a uniform permutation chosen from the family maps any distinct $k$ elements to any distinct $k$ elements equally likely. Efficient constructions of $k$-wise independent permutations are known for $k=2$ and $k=3$, but are unknown for $k \ge 4$. In fact, ... more >>>


TR09-120 | 18th November 2009
Charanjit Jutla

Almost Optimal Bounds for Direct Product Threshold Theorem

Revisions: 2

We consider weakly-verifiable puzzles which are challenge-response puzzles such that the responder may not
be able to verify for itself whether it answered the challenge correctly. We consider $k$-wise direct product of
such puzzles, where now the responder has to solve $k$ puzzles chosen independently in parallel.
Canetti et ... more >>>


TR10-183 | 29th November 2010
Raghu Meka

Almost Optimal Explicit Johnson-Lindenstrauss Transformations

Revisions: 2

The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the Johnson-Lindenstrauss property involve randomness. We address the question of explicitly constructing such embedding families and provide a construction ... more >>>


TR03-066 | 2nd September 2003
Daniele Micciancio

Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>


TR15-200 | 4th December 2015
Andris Ambainis

Almost quadratic gap between partition complexity and query/communication complexity

Revisions: 1

We show nearly quadratic separations between two pairs of complexity measures:
1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$;
2. As a consequence, we obtain that there is ... more >>>


TR16-195 | 19th November 2016
Pasin Manurangsi

Almost-Polynomial Ratio ETH-Hardness of Approximating Densest $k$-Subgraph

In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., ... more >>>


TR14-012 | 27th January 2014
Scott Aaronson, Russell Impagliazzo, Dana Moshkovitz

AM with Multiple Merlins

Revisions: 1

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close ... more >>>


TR16-054 | 11th April 2016
Omri Weinstein, Huacheng Yu

Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication

This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic
data structure problems. We introduce a new randomized nondeterministic four-party communication model
that enables "accelerated", error-preserving simulations of dynamic data structures.

We use this technique to prove an $\Omega(n\left(\log n/\log\log n\right)^2)$ cell-probe ... more >>>


TR15-158 | 27th September 2015
Ofer Grossman, Dana Moshkovitz

Amplification and Derandomization Without Slowdown

We present techniques for decreasing the error probability of randomized algorithms and for converting randomized algorithms to deterministic (non-uniform) algorithms. Unlike most existing techniques that involve repetition of the randomized algorithm, and hence a slowdown, our techniques produce algorithms with a similar run-time to the original randomized algorithms.

The ... more >>>


TR15-031 | 2nd March 2015
Marco Molinaro, David Woodruff, Grigory Yaroslavtsev

Amplification of One-Way Information Complexity via Codes and Noise Sensitivity

Revisions: 1

We show a new connection between the information complexity of one-way communication problems under product distributions and a relaxed notion of list-decodable codes. As a consequence, we obtain a characterization of the information complexity of one-way problems under product distributions for any error rate based on covering numbers. This generalizes ... more >>>


TR08-038 | 4th April 2008
Eric Allender, Michal Koucky

Amplifying Lower Bounds by Means of Self-Reducibility

Revisions: 2

We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the ... more >>>



TR15-102 | 22nd June 2015
Mario Szegedy

An $O(n^{0.4732})$ upper bound on the complexity of the GKS communication game

We give an $5\cdot n^{\log_{30}5}$ upper bund on the complexity of the communication game introduced by G. Gilmer, M. Kouck\'y and M. Saks \cite{saks} to study the Sensitivity Conjecture \cite{linial}, improving on their
$\sqrt{999\over 1000}\sqrt{n}$ bound. We also determine the exact complexity of the game up to $n\le 9$.
more >>>


TR15-016 | 16th January 2015
Diptarka Chakraborty, Raghunath Tewari

An $O(n^{\epsilon})$ Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

Revisions: 1

Given a graph $G$ and two vertices $s$ and $t$ in it, {\em graph reachability} is the problem of checking whether there exists a path from $s$ to $t$ in $G$. We show that reachability in directed layered planar graphs can be decided in polynomial time and $O(n^\epsilon)$ space, for ... more >>>


TR16-126 | 8th August 2016
Subhash Khot, Igor Shinkar

An $\widetilde{O}(n)$ Queries Adaptive Tester for Unateness

We present an adaptive tester for the unateness property of Boolean functions. Given a function $f:\{0,1\}^n \to \{0,1\}$ the tester makes $O(n \log(n)/\epsilon)$ adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 any function that is $\epsilon$-far from being unate.
more >>>


TR17-029 | 18th February 2017
Clement Canonne, Tom Gur

An Adaptivity Hierarchy Theorem for Property Testing

Revisions: 1

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all ... more >>>


TR11-157 | 25th November 2011
Eli Ben-Sasson, Shachar Lovett, Noga Ron-Zewi

An additive combinatorics approach to the log-rank conjecture in communication complexity

Revisions: 1

For a {0,1}-valued matrix $M$ let CC($M$) denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovasz and Saks [FOCS 1988] states that CC($M$) is at most $\log^c({\mbox{rank}}(M))$ for some absolute constant $c$ where rank($M$) denotes the rank of $M$ over the field ... more >>>


TR96-010 | 9th February 1996
Christoph Meinel, Anna Slobodova

An Adequate Reducibility Concept for Problems Defined in Terms of Ordered Binary Decision Diagrams

Revisions: 1

Reducibility concepts are fundamental in complexity theory.
Usually, they are defined as follows: A problem P is reducible
to a problem S if P can be computed using a program or device
for S as a subroutine. However, in the case of such restricted
models as ... more >>>


TR11-172 | 20th December 2011
Yang Cai, Constantinos Daskalakis, S. Matthew Weinberg

An Algorithmic Characterization of Multi-Dimensional Mechanisms

We obtain a characterization of feasible, Bayesian, multi-item multi-bidder mechanisms with independent, additive bidders as distributions over hierarchical mechanisms. Combined with cyclic-monotonicity our results provide a complete characterization of feasible, Bayesian Incentive Compatible mechanisms for this setting.

Our characterization is enabled by a novel, constructive proof of Border's theorem [Border ... more >>>


TR16-143 | 15th September 2016
Nikhil Balaji, Nutan Limaye, Srikanth Srinivasan

An Almost Cubic Lower Bound for $\Sigma\Pi\Sigma$ Circuits Computing a Polynomial in VP

In this note, we prove that there is an explicit polynomial in VP such that any $\Sigma\Pi\Sigma$ arithmetic circuit computing it must have size at least $n^{3-o(1)}$. Up to $n^{o(1)}$ factors, this strengthens a recent result of Kayal, Saha and Tavenas (ICALP 2016) which gives a polynomial in VNP with ... more >>>


TR16-006 | 22nd January 2016
Neeraj Kayal, Chandan Saha, Sébastien Tavenas

An almost Cubic Lower Bound for Depth Three Arithmetic Circuits

Revisions: 2

We show an $\Omega \left(\frac{n^3}{(\ln n)^2}\right)$ lower bound on the size of any depth three ($\SPS$) arithmetic circuit computing an explicit multilinear polynomial in $n$ variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson.

more >>>

TR08-108 | 19th November 2008
Nitin Saxena, C. Seshadhri

An Almost Optimal Rank Bound for Depth-3 Identities

We show that the rank of a depth-3 circuit (over any field) that is simple,
minimal and zero is at most O(k^3\log d). The previous best rank bound known was
2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).
This almost resolves the rank question first posed by ... more >>>


TR14-114 | 27th August 2014
Roei Tell

An Alternative Proof of an $\Omega(k)$ Lower Bound for Testing $k$-linear Boolean Functions

We provide an alternative proof for a known result stating that $\Omega(k)$ queries are needed to test $k$-sparse linear Boolean functions. Similar to the approach of Blais and Kane (2012), we reduce the proof to the analysis of Hamming weights of vectors in affi ne subspaces of the Boolean hypercube. ... more >>>


TR10-096 | 16th June 2010
Dana Moshkovitz

An Alternative Proof of The Schwartz-Zippel Lemma

Revisions: 1

We show a non-inductive proof of the Schwartz-Zippel lemma. The lemma bounds the number of zeros of a multivariate low degree polynomial over a finite field.

more >>>

TR00-018 | 16th February 2000
Oliver Kullmann

An application of matroid theory to the SAT problem

A basic property of minimally unsatisfiable clause-sets F is that
c(F) >= n(F) + 1 holds, where c(F) is the number of clauses, and
n(F) the number of variables. Let MUSAT(k) be the class of minimally
unsatisfiable clause-sets F with c(F) <= n(F) + k.

Poly-time decision algorithms are known ... more >>>


TR04-110 | 24th November 2004
Tomoyuki Yamakami, Harumichi Nishimura

An Application of Quantum Finite Automata to Interactive Proof Systems

Quantum finite automata have been studied intensively since
their introduction in late 1990s as a natural model of a
quantum computer with finite-dimensional quantum memory space.
This paper seeks their direct application
to interactive proof systems in which a mighty quantum prover
communicates with a ... more >>>


TR09-085 | 14th September 2009
Christoph Behle, Andreas Krebs, Stephanie Reifferscheid

An Approach to characterize the Regular Languages in TC0 with Linear Wires

Revisions: 1

We consider the regular languages recognized by weighted threshold circuits with a linear number of wires.
We present a simple proof to show that parity cannot be computed by such circuits.
Our proofs are based on an explicit construction to restrict the input of the circuit such that the value ... more >>>


TR14-030 | 5th March 2014
Dana Moshkovitz

An Approach To The Sliding Scale Conjecture Via Parallel Repetition For Low Degree Testing

The Sliding Scale Conjecture in PCP states that there are PCP verifiers with a constant number of queries and soundness error that is exponentially small in the randomness of the verifier and the length of the prover's answers.

The Sliding Scale Conjecture is one of the oldest open problems in ... more >>>


TR97-017 | 5th May 1997
Marek Karpinski, Juergen Wirtgen, Alexander Zelikovsky

An Approximation Algorithm for the Bandwidth Problem on Dense Graphs

The bandwidth problem is the problem of numbering the vertices of a
given graph $G$ such that the maximum difference between the numbers
of adjacent vertices is minimal. The problem has a long history and
is known to be NP-complete Papadimitriou [Pa76]. Only few special
cases ... more >>>


TR04-048 | 21st April 2004
André Lanka, Andreas Goerdt

An approximation hardness result for bipartite Clique

Assuming 3-SAT formulas are hard to refute
on average, Feige showed some approximation hardness
results for several problems like min bisection, dense
$k$-subgraph, max bipartite clique and the 2-catalog segmentation
problem. We show a similar result for
max bipartite clique, but under the assumption, 4-SAT formulas
are hard to refute ... more >>>


TR15-065 | 18th April 2015
Benjamin Rossman, Rocco Servedio, Li-Yang Tan

An average-case depth hierarchy theorem for Boolean circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ ... more >>>


TR16-041 | 17th March 2016
Johan Hastad

An average-case depth hierarchy theorem for higher depth

We extend the recent hierarchy results of Rossman, Servedio and
Tan \cite{rst15} to any $d \leq \frac {c \log n}{\log {\log n}}$
for an explicit constant $c$.

To be more precise, we prove that for any such $d$ there is a function
$F_d$ that is computable by a read-once formula ... more >>>


TR98-069 | 7th December 1998
Rüdiger Reischuk, Thomas Zeugmann

An Average-Case Optimal One-Variable Pattern Language Learner


A new algorithm for learning one-variable pattern languages from positive data
is proposed and analyzed with respect to its average-case behavior.
We consider the total learning time that takes into account all
operations till convergence to a correct hypothesis is achieved.

For almost all meaningful distributions
defining how ... more >>>


TR95-038 | 2nd July 1995
Joe Kilian, Erez Petrank

An Efficient Non-Interactive Zero-Knowledge Proof System for NP with General Assumptions

We consider noninteractive zero-knowledge proofs in the shared random
string model proposed by Blum, Feldman and Micali \cite{bfm}. Until
recently there was a sizable polynomial gap between the most
efficient noninteractive proofs for {\sf NP} based on general
complexity assumptions \cite{fls} versus those based on specific
algebraic assumptions \cite{Da}. ... more >>>


TR06-119 | 13th September 2006
Noga Alon, Oded Schwartz, Asaf Shapira

An Elementary Construction of Constant-Degree Expanders

We describe a short and easy to analyze construction of
constant-degree expanders. The construction relies on the
replacement-product, which we analyze using an elementary
combinatorial argument. The construction applies the replacement
product (only twice!) to turn the Cayley expanders of \cite{AR},
whose degree is polylog n, into constant degree
expanders.

... more >>>

TR11-026 | 27th February 2011
Evgeny Demenkov, Alexander Kulikov

An Elementary Proof of $3n-o(n)$ Lower Bound on the Circuit Complexity of Affine Dispersers

A Boolean function $f \colon \mathbb{F}^n_2 \rightarrow \mathbb{F}_2$ is called an affine disperser for sources of dimension $d$, if $f$ is not constant on any affine subspace of $\mathbb{F}^n_2$ of dimension at least $d$. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for $d=o(n)$. The main ... more >>>


TR10-182 | 26th November 2010
Shachar Lovett

An elementary proof of anti-concentration of polynomials in Gaussian variables

Recently there has been much interest in polynomial threshold functions in the context of learning theory, structural results and pseudorandomness. A crucial ingredient in these works is the understanding of the distribution of low-degree multivariate polynomials evaluated over normally distributed inputs. In particular, the two important properties are exponential tail ... more >>>


TR10-091 | 14th May 2010
Nikolay Vereshchagin

An Encoding Invariant Version of Polynomial Time Computable Distributions

When we represent a decision problem,like CIRCUIT-SAT, as a language over the binary alphabet,
we usually do not specify how to encode instances by binary strings.
This relies on the empirical observation that the truth of a statement of the form ``CIRCUIT-SAT belongs to a complexity class $C$''
more >>>


TR14-165 | 3rd December 2014
Venkatesan Guruswami, Ameya Velingker

An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

We prove a lower estimate on the increase in entropy when two copies of a conditional random variable $X | Y$, with $X$ supported on $\mathbb{Z}_q=\{0,1,\dots,q-1\}$ for prime $q$, are summed modulo $q$. Specifically, given two i.i.d. copies $(X_1,Y_1)$ and $(X_2,Y_2)$ of a pair of random variables $(X,Y)$, with $X$ ... more >>>


TR01-083 | 29th October 2001
Nikolay Vereshchagin

An enumerable undecidable set with low prefix complexity: a simplified proof

Revisions: 1

We present a simplified proof of Solovay-Calude-Coles theorem
stating that there is an enumerable undecidable set with the
following property: prefix
complexity of its initial segment of length n is bounded by prefix
complexity of n (up to an additive constant).

more >>>

TR03-013 | 7th March 2003
Luca Trevisan

An epsilon-Biased Generator in NC0

Comments: 1

Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there ... more >>>


TR12-127 | 3rd October 2012
Eshan Chattopadhyay, Adam Klivans, Pravesh Kothari

An Explicit VC-Theorem for Low-Degree Polynomials

Let $X \subseteq \mathbb{R}^{n}$ and let ${\mathcal C}$ be a class of functions mapping $\mathbb{R}^{n} \rightarrow \{-1,1\}.$ The famous VC-Theorem states that a random subset $S$ of $X$ of size $O(\frac{d}{\epsilon^{2}} \log \frac{d}{\epsilon})$, where $d$ is the VC-Dimension of ${\mathcal C}$, is (with constant probability) an $\epsilon$-approximation for ${\mathcal C}$ ... more >>>


TR07-007 | 17th January 2007
Jan Krajicek

An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams

We prove an exponential lower bound on the size of proofs
in the proof system operating with ordered binary decision diagrams
introduced by Atserias, Kolaitis and Vardi. In fact, the lower bound
applies to semantic derivations operating with sets defined by OBDDs.
We do not assume ... more >>>


TR12-098 | 3rd August 2012
Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

An exponential lower bound for homogeneous depth four arithmetic circuits with bounded bottom fanin

Revisions: 3

Agrawal and Vinay (FOCS 2008) have recently shown that an exponential lower bound for depth four homogeneous circuits with bottom layer of $\times$ gates having sublinear fanin translates to an exponential lower bound for a general arithmetic circuit computing the permanent. Motivated by this, we examine the complexity of computing ... more >>>


TR14-005 | 14th January 2014
Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas. That is, we give
an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with $0, 1$-coefficients such that
for any representation of ... more >>>


TR15-109 | 1st July 2015
Mrinal Kumar, Ramprasad Saptharishi

An exponential lower bound for homogeneous depth-5 circuits over finite fields

In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in N\}$ of polynomials in $VNP$, where $P_d$ is of degree $d$ in $n = d^{O(1)}$ variables, ... more >>>


TR12-081 | 26th June 2012
Neeraj Kayal

An exponential lower bound for the sum of powers of bounded degree polynomials

Revisions: 1

In this work we consider representations of multivariate polynomials in $F[x]$ of the form $ f(x) = Q_1(x)^{e_1} + Q_2(x)^{e_2} + ... + Q_s(x)^{e_s},$ where the $e_i$'s are positive integers and the $Q_i$'s are arbitary multivariate polynomials of bounded degree. We give an explicit $n$-variate polynomial $f$ of degree $n$ ... more >>>


TR95-057 | 24th November 1995
Dima Grigoriev, Marek Karpinski, A. C. Yao

An Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX

We prove an exponential lower bound on the size of any
fixed-degree algebraic decision tree for solving MAX, the
problem of finding the maximum of $n$ real numbers. This
complements the $n-1$ lower bound of Rabin \cite{R72} on
the depth of ... more >>>


TR94-018 | 12th December 1994
Jan Krajicek, Pavel Pudlak, Alan Woods

An Exponential Lower Bound to the Size of Bounded Depth Frege Proofs of the Pigeonhole Principle

We prove lower bounds of the form $exp\left(n^{\epsilon_d}\right),$
$\epsilon_d>0,$ on the length of proofs of an explicit sequence of
tautologies, based on the Pigeonhole Principle, in proof systems
using formulas of depth $d,$ for any constant $d.$ This is the
largest lower bound for the strongest proof system, for which ... more >>>


TR01-056 | 6th August 2001
Michael Alekhnovich, Jan Johannsen, Alasdair Urquhart

An Exponential Separation between Regular and General Resolution

This paper gives two distinct proofs of an exponential separation
between regular resolution and unrestricted resolution.
The previous best known separation between these systems was
quasi-polynomial.

more >>>

TR15-173 | 29th October 2015
Martin Schwarz

An exponential time upper bound for Quantum Merlin-Arthur games with unentangled provers

We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce ... more >>>


TR07-046 | 23rd April 2007
Philipp Hertel

An Exponential Time/Space Speedup For Resolution

Comments: 1

Satisfiability algorithms have become one of the most practical and successful approaches for solving a variety of real-world problems, including hardware verification, experimental design, planning and diagnosis problems. The main reason for the success is due to highly optimized algorithms for SAT based on resolution. The most successful of these ... more >>>


TR07-034 | 29th March 2007
Anup Rao

An Exposition of Bourgain's 2-Source Extractor

A construction of Bourgain gave the first 2-source
extractor to break the min-entropy rate 1/2 barrier. In this note,
we write an exposition of his result, giving a high level way to view
his extractor construction.

We also include a proof of a generalization of Vazirani's XOR lemma
that seems ... more >>>


TR12-029 | 3rd April 2012
Shachar Lovett

An exposition of Sanders quasi-polynomial Freiman-Ruzsa theorem

The polynomial Freiman-Ruzsa conjecture is one of the important conjectures in additive combinatorics. It asserts than one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying parameters. This conjecture has also already found several applications in theoretical computer science. Recently, Tom ... more >>>


TR05-067 | 28th June 2005
Zeev Dvir, Amir Shpilka

An Improved Analysis of Mergers

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>


TR06-107 | 26th August 2006
Arkadev Chattopadhyay

An improved bound on correlation between polynomials over Z_m and MOD_q

Revisions: 1

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>


TR15-117 | 21st July 2015
Boris Bukh, Venkatesan Guruswami

An improved bound on the fraction of correctable deletions

Revisions: 1

We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed $k \ge 2$, we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching $1-\frac{2}{k+1}$. In particular, for binary codes, we are able to ... more >>>


TR13-150 | 4th November 2013
Ruiwen Chen, Valentine Kabanets, Nitin Saurabh

An Improved Deterministic #SAT Algorithm for Small De Morgan Formulas

We give a deterministic #SAT algorithm for de Morgan formulas of size up to $n^{2.63}$, which runs in time $2^{n-n^{\Omega(1)}}$. This improves upon the deterministic #SAT algorithm of \cite{CKKSZ13}, which has similar running time but works only for formulas of size less than $n^{2.5}$.

Our new algorithm is based on ... more >>>


TR17-030 | 15th February 2017
Amey Bhangale, Subhash Khot, Devanathan Thiruvenkatachari

An Improved Dictatorship Test with Perfect Completeness

A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.

... more >>>

TR00-057 | 25th July 2000
Martin Sauerhoff

An Improved Hierarchy Result for Partitioned BDDs

One of the great challenges of complexity theory is the problem of
analyzing the dependence of the complexity of Boolean functions on the
resources nondeterminism and randomness. So far, this problem could be
solved only for very few models of computation. For so-called
partitioned binary decision diagrams, which are a ... more >>>


TR16-206 | 24th December 2016
Benjamin Rossman

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [39] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC$^0$ ... more >>>


TR12-099 | 5th August 2012
Nikos Leonardos

An improved lower bound for the randomized decision tree complexity of recursive majority

Revisions: 1

We prove that the randomized decision tree complexity of the recursive majority-of-three is $\Omega(2.6^d)$, where $d$ is the depth of the recursion. The proof is by a bottom up induction, which is same in spirit as the one in the proof of Saks and Wigderson in their FOCS 1986 paper ... more >>>


TR05-030 | 12th February 2005
Evgeny Dantsin, Alexander Wolpert

An Improved Upper Bound for SAT

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. Its running time is at most $2^{n(1-1/\alpha)}$ up to a polynomial factor, where $\alpha = \ln(m/n) + O(\ln \ln m)$ and $n$, $m$ are respectively the number of variables ... more >>>


TR07-117 | 8th November 2007
Edward Hirsch, Dmitry Itsykson

An infinitely-often one-way function based on an average-case assumption

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, ... more >>>


TR12-131 | 18th October 2012
Mark Braverman, Ankur Moitra

An Information Complexity Approach to Extended Formulations

Revisions: 1

We prove an unconditional lower bound that any linear program that achieves an $O(n^{1-\epsilon})$ approximation for clique has size $2^{\Omega(n^\epsilon)}$. There has been considerable recent interest in proving unconditional lower bounds against any linear program. Fiorini et al proved that there is no polynomial sized linear program for traveling salesman. ... more >>>


TR14-047 | 8th April 2014
Mark Braverman, Omri Weinstein

An Interactive Information Odometer with Applications

Revisions: 1

We introduce a novel technique which enables two players to maintain an estimate of the internal information cost of their conversation in an online fashion without revealing much extra information. We use this construction to obtain new results about communication complexity and information-theoretically secure computation.

As a first corollary, ... more >>>


TR09-144 | 24th December 2009
Prahladh Harsha, Adam Klivans, Raghu Meka

An Invariance Principle for Polytopes

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$
formed by the intersection of $k$ halfspaces, we prove that
$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ... more >>>


TR06-011 | 2nd January 2006
Yijia Chen, Martin Grohe

An Isomorphism between Subexponential and Parameterized Complexity Theory

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

more >>>

TR96-002 | 10th January 1996
Manindra Agrawal, Eric Allender

An Isomorphism Theorem for Circuit Complexity

We show that all sets complete for NC$^1$ under AC$^0$
reductions are isomorphic under AC$^0$-computable isomorphisms.

Although our proof does not generalize directly to other
complexity classes, we do show that, for all complexity classes C
closed under NC$^1$-computable many-one reductions, the sets ... more >>>


TR04-114 | 21st November 2004
Vladimir Trifonov

An O(log n log log n) Space Algorithm for Undirected s,t-Connectivity

We present a deterministic O(log n log log n) space algorithm for
undirected s,t-connectivity. It is based on the deterministic EREW
algorithm of Chong and Lam (SODA 93) and uses the universal
exploration sequences for trees constructed by Kouck\'y (CCC 01).
Our result improves the O(log^{4/3} n) bound of Armoni ... more >>>


TR06-050 | 18th April 2006
Alexander Razborov, Sergey Yekhanin

An Omega(n^{1/3}) Lower Bound for Bilinear Group Based Private Information Retrieval

A two server private information retrieval (PIR) scheme
allows a user U to retrieve the i-th bit of an
n-bit string x replicated between two servers while each
server individually learns no information about i. The main
parameter of interest in a PIR scheme is its communication
complexity, namely the ... more >>>


TR13-062 | 18th April 2013
Deeparnab Chakrabarty, C. Seshadhri

An optimal lower bound for monotonicity testing over hypergrids

For positive integers $n, d$, consider the hypergrid $[n]^d$ with the coordinate-wise product partial ordering denoted by $\prec$.
A function $f: [n]^d \mapsto \mathbb{N}$ is monotone if $\forall x \prec y$, $f(x) \leq f(y)$.
A function $f$ is $\varepsilon$-far from monotone if at least an $\varepsilon$-fraction of values must ... more >>>


TR10-140 | 17th September 2010
Amit Chakrabarti, Oded Regev

An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

We prove an optimal $\Omega(n)$ lower bound on the randomized
communication complexity of the much-studied
Gap-Hamming-Distance problem. As a consequence, we
obtain essentially optimal multi-pass space lower bounds in the
data stream model for a number of fundamental problems, including
the estimation of frequency moments.

The Gap-Hamming-Distance problem is a ... more >>>


TR03-070 | 19th August 2003
Amit Chakrabarti, Oded Regev

An Optimal Randomised Cell Probe Lower Bound for Approximate Nearest Neighbour Searching

We consider the approximate nearest neighbour search problem on the
Hamming Cube $\b^d$. We show that a randomised cell probe algorithm that
uses polynomial storage and word size $d^{O(1)}$ requires a worst case
query time of $\Omega(\log\log d/\log\log\log d)$. The approximation
factor may be as loose as $2^{\log^{1-\eta}d}$ for any ... more >>>


TR96-020 | 6th March 1996
C.P. Schnorr, Carsten Rössner

An Optimal, Stable Continued Fraction Algorithm for Arbitrary Dimension


TR15-130 | 11th August 2015
Ronald de Haan

An Overview of Non-Uniform Parameterized Complexity

We consider several non-uniform variants of parameterized complexity classes that have been considered in the literature. We do so in a homogenous notation, allowing a clear comparison of the various variants. Additionally, we consider some novel (non-uniform) parameterized complexity classes that come up in the framework of parameterized knowledge compilation. ... more >>>


TR15-033 | 6th March 2015
Alexander Razborov

An Ultimate Trade-Off in Propositional Proof Complexity

Revisions: 1

We exhibit an unusually strong trade-off between resolution proof width and tree-like proof size. Namely, we show that for any parameter $k=k(n)$ there are unsatisfiable $k$-CNFs that possess refutations of width $O(k)$, but such that any tree-like refutation of width $n^{1-\epsilon}/k$ must necessarily have {\em double} exponential size $\exp(n^{\Omega(k)})$. Conceptually, ... more >>>


TR06-056 | 27th April 2006
Salil Vadhan

An Unconditional Study of Computational Zero Knowledge

We prove a number of general theorems about ZK, the class of problems possessing (computational) zero-knowledge proofs. Our results are unconditional, in contrast to most previous works on ZK, which rely on the assumption that one-way functions exist.

We establish several new characterizations of ZK, and use these characterizations to ... more >>>


TR95-010 | 16th February 1995
Pavel Pudlak, Jiri Sgall

An Upper Bound for a Communication Game Related to Time-Space Tradeoffs


We prove an unexpected upper bound on a communication game proposed
by Jeff Edmonds and Russell Impagliazzo as an approach for
proving lower bounds for time-space tradeoffs for branching programs.
Our result is based on a generalization of a construction of Erdos,
Frankl and Rodl of a large 3-hypergraph ... more >>>


TR01-079 | 6th September 2001
Michele Zito

An Upper Bound on the Space Complexity of Random Formulae in Resolution

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

more >>>

TR97-052 | 11th November 1997
Eduardo D. Sontag

Analog Neural Nets with Gaussian or other Common Noise Distributions cannot Recognize Arbitrary Regular Languages

We consider recurrent analog neural nets where the output of each
gate is subject to Gaussian noise, or any other common noise
distribution that is nonzero on a large set.
We show that many regular languages cannot be recognized by
networks of this type, and
more >>>


TR95-025 | 8th May 1995
Günter Hotz, Gero Vierke, Bjoern Schieffer

Analytic Machines

Comments: 1

In this paper the $R$-machines defined by Blum, Shub and Smale
are generalized by allowing infinite convergent computations.
The description of real numbers is infinite.
Therefore, considering arithmetic operations on real numbers should
also imply infinite computations on {\em analytic machines}.
We prove that $\R$-computable functions are $\Q$-analytic.
We show ... more >>>


TR05-025 | 20th February 2005
Zeev Dvir, Ran Raz

Analyzing Linear Mergers

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>


TR12-022 | 14th March 2012
Amit Chakrabarti, Graham Cormode, Andrew McGregor, Justin Thaler

Annotations in Data Streams

Revisions: 1

The central goal of data stream algorithms is to process massive streams of data using sublinear storage space. Motivated by work in the database community on outsourcing database and data stream processing, we ask whether the space usage of such algorithms can be further reduced by enlisting a more powerful ... more >>>


TR97-045 | 29th September 1997
Oded Goldreich, David Zuckerman

Another proof that BPP subseteq PH (and more).

Comments: 1


We provide another proof of the Sipser--Lautemann Theorem
by which $BPP\subseteq MA$ ($\subseteq PH$).
The current proof is based on strong
results regarding the amplification of $BPP$, due to Zuckerman.
Given these results, the current proof is even simpler than previous ones.
Furthermore, extending the proof leads ... more >>>


TR06-143 | 15th November 2006
Frank Neumann, Carsten Witt

Ant Colony Optimization and the Minimum Spanning Tree Problem

Ant Colony Optimization (ACO) is a kind of randomized search heuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. ... more >>>


TR13-147 | 25th October 2013
Adam Bouland, Scott Aaronson

Any Beamsplitter Generates Universal Quantum Linear Optics

Revisions: 3

In 1994, Reck et al. showed how to realize any linear-optical unitary transformation using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of m by m unitary transformations (or orthogonal transformations, in the real case) ... more >>>


TR14-061 | 21st April 2014
Raghav Kulkarni, Youming Qiao, Xiaoming Sun

Any Monotone Property of 3-uniform Hypergraphs is Weakly Evasive

For a Boolean function $f,$ let $D(f)$ denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine $f.$ In a classic paper,
Rivest and Vuillemin \cite{rv} show that any non-constant monotone property $\mathcal{P} : \{0, 1\}^{n \choose 2} \to ... more >>>


TR00-052 | 3rd July 2000
Beate Bollig, Ingo Wegener

Approximability and Nonapproximability by Binary Decision Diagrams

Many BDD (binary decision diagram) models are motivated
by CAD applications and have led to complexity theoretical
problems and results. Motivated by applications in genetic
programming Krause, Savick\'y, and Wegener (1999) have shown
that for the inner product function IP$_n$ and the direct
storage access function DSA$_n$ ... more >>>


TR00-091 | 21st December 2000
Cristina Bazgan, Wenceslas Fernandez de la Vega, Marek Karpinski

Approximability of Dense Instances of NEAREST CODEWORD Problem

We give a polynomial time approximation scheme (PTAS) for dense
instances of the NEAREST CODEWORD problem.

more >>>

TR03-056 | 29th July 2003
Piotr Berman, Marek Karpinski

Approximability of Hypergraph Minimum Bisection

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>


TR06-045 | 13th March 2006
Jan Arpe, Bodo Manthey

Approximability of Minimum AND-Circuits

Revisions: 1

Given a set of monomials, the Minimum AND-Circuit problem asks for a
circuit that computes these monomials using AND-gates of fan-in two and
being of minimum size. We prove that the problem is not polynomial time
approximable within a factor of less than 1.0051 unless P = NP, even if
more >>>


TR14-065 | 2nd May 2014
Andrzej Dudek , Marek Karpinski, Andrzej Rucinski, Edyta Szymanska

Approximate Counting of Matchings in $(3,3)$-Hypergraphs

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting ... more >>>


TR02-031 | 30th April 2002
Vikraman Arvind, Venkatesh Raman

Approximate Counting small subgraphs of bounded treewidth and related problems

Revisions: 1

We give a randomized approximation algorithm taking
$O(k^{O(k)}n^{b+O(1)})$ time to count the number of copies of a
$k$-vertex graph with treewidth at most $b$ in an $n$ vertex graph
$G$ with approximation ratio $1/k^{O(k)}$ and error probability
inverse exponential in $n$. This algorithm is based on ... more >>>


TR16-121 | 4th August 2016
Mark Bun, Justin Thaler

Approximate Degree and the Complexity of Depth Three Circuits

Revisions: 1

Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the ... more >>>


TR12-078 | 14th June 2012
Vikraman Arvind, Sebastian Kuhnert, Johannes Köbler, Yadu Vasudev

Approximate Graph Isomorphism

We study optimization versions of Graph Isomorphism. Given two graphs $G_1,G_2$, we are interested in finding a bijection $\pi$ from $V(G_1)$ to $V(G_2)$ that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an $n^{O(\log n)}$ time approximation scheme that for any constant ... more >>>


TR07-116 | 25th September 2007
Alexander A. Sherstov

Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Comments: 1

Let A_1,...,A_n be events in a probability space. The
approximate inclusion-exclusion problem, due to Linial and
Nisan (1990), is to estimate Pr[A_1 OR ... OR A_n] given
Pr[AND_{i\in S}A_i] for all |S|<=k. Kahn et al. (1996) solve
this problem optimally for each k. We study the following more
general question: ... more >>>


TR14-046 | 8th April 2014
Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff

Approximate Nonnegative Rank is Equivalent to the Smooth Rectangle Bound

We consider two known lower bounds on randomized communication complexity: The smooth rectangle bound and the logarithm of the approximate non-negative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term.
The logarithm of the nonnegative rank is known to ... more >>>


TR10-032 | 19th January 2010
Jack H. Lutz, Brad Shutters

Approximate Self-Assembly of the Sierpinski Triangle

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in ... more >>>


TR15-046 | 2nd April 2015
Talya Eden, Amit Levi, Dana Ron

Approximately Counting Triangles in Sublinear Time

Comments: 1

We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in two models: Exact counting algorithms, which require reading the entire graph, and streaming algorithms, where the edges are given in a stream and the memory is limited. In this work ... more >>>


TR11-171 | 15th December 2011
Piotr Indyk, Reut Levi, Ronitt Rubinfeld

Approximating and Testing $k$-Histogram Distributions in Sub-linear time

Revisions: 1

A discrete distribution $p$, over $[n]$, is a $k$-histogram if its probability distribution function can be
represented as a piece-wise constant function with $k$ pieces. Such a function
is
represented by a list of $k$ intervals and $k$ corresponding values. We consider
the following problem: given a collection of samples ... more >>>


TR05-073 | 14th July 2005
Oded Goldreich, Dana Ron

Approximating Average Parameters of Graphs.


Inspired by Feige ({\em 36th STOC}, 2004),
we initiate a study of sublinear randomized algorithms
for approximating average parameters of a graph.
Specifically, we consider the average degree of a graph
and the average distance between pairs of vertices in a graph.
Since our focus is on sublinear algorithms, ... more >>>


TR13-051 | 2nd April 2013
Eric Blais, Li-Yang Tan

Approximating Boolean functions with depth-2 circuits

We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function.
In the first direction, our main positive results are the first non-trivial universal upper bounds on ... more >>>


TR01-042 | 31st May 2001
Marek Karpinski

Approximating Bounded Degree Instances of NP-Hard Problems

We present some of the recent results on computational complexity
of approximating bounded degree combinatorial optimization problems. In
particular, we present the best up to now known explicit nonapproximability
bounds on the very small degree optimization problems which are of
particular importance on the intermediate stages ... more >>>


TR12-074 | 12th June 2012
Venkatesan Guruswami, Yuan Zhou

Approximating Bounded Occurrence Ordering CSPs

A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most $O(1)$ constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive ... more >>>


TR06-007 | 23rd November 2005
MohammadTaghi Hajiaghayi, Guy Kortsarz, Mohammad R. Salavatipour

Approximating Buy-at-Bulk $k$-Steiner trees

In the buy-at-bulk $k$-Steiner tree (or rent-or-buy
$k$-Steiner tree) problem we are given a graph $G(V,E)$ with a set
of terminals $T\subseteq V$ including a particular vertex $s$ called
the root, and an integer $k\leq |T|$. There are two cost functions
on the edges of $G$, a buy cost $b:E\longrightarrow ... more >>>


TR07-027 | 2nd February 2007
Tobias Friedrich, Jun He, Nils Hebbinghaus, Frank Neumann, Carsten Witt

Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical results on this subject.
We consider the approximation ability of randomized search for the class of ... more >>>


TR16-116 | 26th July 2016
Subhash Khot, Rishi Saket

Approximating CSPs using LP Relaxation

This paper studies how well the standard LP relaxation approximates a $k$-ary constraint satisfaction problem (CSP) on label set $[L]$. We show that, assuming the Unique Games Conjecture, it achieves an approximation within $O(k^3\cdot \log L)$ of the optimal approximation factor. In particular we prove the following hardness result: let ... more >>>


TR98-048 | 6th July 1998
Irit Dinur, Guy Kindler, Shmuel Safra

Approximating CVP to Within Almost Polynomial Factor is NP-Hard

This paper shows finding the closest vector in a lattice
to be NP-hard to approximate to within any factor up to
$2^{(\log{n})^{1-\epsilon}}$ where
$\epsilon = (\log\log{n})^{-\alpha}$
and $\alpha$ is any positive constant $<{1\over 2}$.

more >>>

TR97-004 | 19th February 1997
Marek Karpinski, Alexander Zelikovsky

Approximating Dense Cases of Covering Problems

Comments: 1

We study dense instances of several covering problems. An instance of
the set cover problem with $m$ sets is dense if there is $\epsilon>0$
such that any element belongs to at least $\epsilon m$ sets. We show
that the dense set cover problem can be approximated with ... more >>>


TR02-018 | 22nd March 2002
Piotr Berman, Marek Karpinski, Yakov Nekrich

Approximating Huffman Codes in Parallel

In this paper we present some new results on the approximate parallel
construction of Huffman codes. Our algorithm achieves linear work
and logarithmic time, provided that the initial set of elements
is sorted. This is the first parallel algorithm for that problem
with the optimal time and ... more >>>


TR00-072 | 14th July 2000
Peter Auer, Philip M. Long, Aravind Srinivasan

Approximating Hyper-Rectangles: Learning and Pseudo-random Sets

The PAC learning of rectangles has been studied because they have
been found experimentally to yield excellent hypotheses for several
applied learning problems. Also, pseudorandom sets for rectangles
have been actively studied recently because (i) they are a subproblem
common to the derandomization of depth-2 (DNF) ... more >>>


TR10-132 | 18th August 2010
Mahdi Cheraghchi, Johan Hastad, Marcus Isaksson, Ola Svensson

Approximating Linear Threshold Predicates

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>


TR03-032 | 16th April 2003
Andreas Björklund, Thore Husfeldt, Sanjeev Khanna

Approximating Longest Directed Path

We investigate the hardness of approximating the longest path and
the longest cycle in directed graphs on $n$ vertices. We show that
neither of these two problems can be polynomial time approximated
within $n^{1-\epsilon}$ for any $\epsilon>0$ unless
$\text{P}=\text{NP}$. In particular, the result holds for
more >>>


TR01-025 | 28th March 2001
Piotr Berman, Marek Karpinski

Approximating Minimum Unsatisfiability of Linear Equations

We consider the following optimization problem:
given a system of m linear equations in n variables over a certain field,
a feasible solution is any assignment of values to the variables, and the
minimized objective function is the number of equations that are not
satisfied. For ... more >>>


TR10-124 | 18th July 2010
Zhixiang Chen, Bin Fu

Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial.
We first prove ... more >>>


TR01-038 | 14th May 2001
Andreas Jakoby, Maciej Liskiewicz, Rüdiger Reischuk

Approximating Schedules for Dynamic Graphs Efficiently

A model for parallel and distributed programs, the dynamic process graph (DPG),
is investigated under graph-theoretic and complexity aspects.
Such graphs embed constructors for parallel programs,
synchronization mechanisms as well as conditional branches.
They are capable of representing all possible executions of a
parallel or distributed program ... more >>>


TR99-002 | 22nd January 1999
Oded Goldreich, Daniele Micciancio, Shmuel Safra and Jean-Pierre Seifert.

Approximating shortest lattice vectors is not harder than approximating closest lattice vectors.

We show that given oracle access to a subroutine which
returns approximate closest vectors in a lattice, one may find in
polynomial-time approximate shortest vectors in a lattice.
The level of approximation is maintained; that is, for any function
$f$, the following holds:
Suppose that the subroutine, on input a ... more >>>


TR99-016 | 25th April 1999
Irit Dinur

Approximating SVP_\infty to within Almost-Polynomial Factors is NP-hard

This paper shows SVP_\infty and CVP_\infty to be NP-hard to approximate
to within any factor up to $n^{1/\log\log n}$. This improves on the
best previous result \cite{ABSS} that showed quasi-NP-hardness for
smaller factors, namely $2^{\log^{1-\epsilon}n}$ for any constant
$\epsilon>0$. We show a direct reduction from SAT to these
problems, that ... more >>>


TR13-023 | 6th February 2013
Alexander A. Sherstov

Approximating the AND-OR Tree

The approximate degree of a Boolean function $f$ is the least degree of a real polynomial
that approximates $f$ within $1/3$ at every point. We prove that the function $\bigwedge_{i=1}^{n}\bigvee_{j=1}^{n}x_{ij}$,
known as the AND-OR tree, has approximate degree $\Omega(n).$ This lower bound is tight
and closes a ... more >>>


TR14-092 | 22nd July 2014
Mark Braverman, Young Kun Ko, Omri Weinstein

Approximating the best Nash Equilibrium in $n^{o(\log n)}$-time breaks the Exponential Time Hypothesis

The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game
initiated a quest for finding \emph{approximate} Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory.

We study the computational complexity of finding an $\eps$-approximate Nash equilibrium with good social ... more >>>


TR05-084 | 31st July 2005
Mickey Brautbar, Alex Samorodnitsky

Approximating the entropy of large alphabets

We consider the problem of approximating the entropy of a discrete distribution P on a domain of size q, given access to n independent samples from the distribution. It is known that n > q is necessary, in general, for a good additive estimate of the entropy. A problem of ... more >>>


TR12-025 | 23rd March 2012
Kord Eickmeyer, Kristoffer Arnsfelt Hansen, Elad Verbin

Approximating the minmax value of 3-player games within a constant is as hard as detecting planted cliques

We consider the problem of approximating the minmax value of a multiplayer game in strategic form. We argue that in 3-player games with 0-1 payoffs, approximating the minmax value within an additive constant smaller than $\xi/2$, where $\xi = \frac{3-\sqrt5}{2} \approx 0.382$, is not possible by a polynomial time algorithm. ... more >>>


TR07-092 | 10th July 2007
Piotr Berman, Bhaskar DasGupta

Approximating the Online Set Multicover Problems Via Randomized Winnowing

In this paper, we consider the weighted online set k-multicover problem. In this problem, we have an universe V of elements, a family SS of subsets of V with a positive real cost for every S\in SS, and a ``coverage factor'' (positive integer) k. A subset \{i_0,i_1,\ldots\ \subseteq V of ... more >>>


TR97-059 | 22nd December 1997
Jin-Yi Cai, Ajay Nerurkar

Approximating the SVP to within a factor $\left(1 + \frac{1}{\mathrm{dim}^\epsilon}\right)$ is NP-hard under randomized reductions

Recently Ajtai showed that
to approximate the shortest lattice vector in the $l_2$-norm within a
factor $(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large
constant $k$, is NP-hard under randomized reductions.
We improve this result to show that
to approximate a shortest lattice vector within a
factor $(1+ \mbox{dim}^{-\epsilon})$, for any
$\epsilon>0$, ... more >>>


TR07-119 | 5th December 2007
Piotr Berman, Bhaskar DasGupta, Marek Karpinski

Approximating Transitive Reductions for Directed Networks

We consider <i>minimum equivalent digraph</i> (<i>directed network</i>) problem (also known as the <i>strong transitive reduction</i>), its maximum optimization variant, and some extensions of those two types of problems. We prove the existence of polynomial time approximation algorithms with ratios 1.5 for all the minimization problems and 2 for all the ... more >>>


TR06-063 | 1st May 2006
Moses Charikar, Konstantin Makarychev, Yury Makarychev

Approximation Algorithm for the Max k-CSP Problem

We present a c.k/2^k approximation algorithm for the Max k-CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which has an approximation guarantee of Omega(k/(2^k log k)). Our approximation guarantee matches the upper bound of Samorodnitsky and Trevisan up ... more >>>


TR00-051 | 14th July 2000
Marek Karpinski, Miroslaw Kowaluk, Andrzej Lingas

Approximation Algorithms for MAX-BISECTION on Low Degree Regular Graphs and Planar Graphs

The max-bisection problem is to find a partition of the vertices of a
graph into two equal size subsets that maximizes the number of edges with
endpoints in both subsets.
We obtain new improved approximation ratios for the max-bisection problem on
the low degree $k$-regular graphs for ... more >>>


TR05-034 | 5th April 2005
Luca Trevisan

Approximation Algorithms for Unique Games

Revisions: 1 , Comments: 1

Khot formulated in 2002 the "Unique Games Conjectures" stating that, for any epsilon > 0, given a system of constraints of a certain form, and the promise that there is an assignment that satisfies a 1-epsilon fraction of constraints, it is intractable to find a solution that satisfies even an ... more >>>


TR06-101 | 22nd August 2006
Wenceslas Fernandez de la Vega, Marek Karpinski

Approximation Complexity of Nondense Instances of MAX-CUT

We prove existence of approximation schemes for instances of MAX-CUT with $\Omega(\frac{n^2}{\Delta})$ edges which work in $2^{O^\thicksim(\frac{\Delta}{\varepsilon^2})}n^{O(1)}$ time. This entails in particular existence of quasi-polynomial approximation schemes (QPTASs) for mildly sparse instances of MAX-CUT with $\Omega(\frac{n^2}{\operatorname{polylog} n})$ edges. The result depends on new sampling method for smoothed linear programs that ... more >>>


TR96-030 | 31st March 1996
Meera Sitharam

Approximation from linear spaces and applications to complexity


We develop an analytic framework based on
linear approximation and point out how a number of complexity
related questions --
on circuit and communication
complexity lower bounds, as well as
pseudorandomness, learnability, and general combinatorics
of Boolean functions --
fit neatly into this framework. ... more >>>


TR03-022 | 11th April 2003
Piotr Berman, Marek Karpinski, Alexander D. Scott

Approximation Hardness and Satisfiability of Bounded Occurrence Instances of SAT

We study approximation hardness and satisfiability of bounded
occurrence uniform instances of SAT. Among other things, we prove
the inapproximability for SAT instances in which every clause has
exactly 3 literals and each variable occurs exactly 4 times,
and display an explicit ... more >>>


TR02-073 | 12th December 2002
Janka Chlebíková, Miroslav Chlebík

Approximation Hardness for Small Occurrence Instances of NP-Hard Problem

The paper contributes to the systematic study (started by Berman and
Karpinski) of explicit approximability lower bounds for small occurrence optimization
problems. We present parametrized reductions for some packing and
covering problems, including 3-Dimensional Matching, and prove the best
known inapproximability results even for highly restricted versions of ... more >>>


TR01-026 | 3rd April 2001
Piotr Berman, Marek Karpinski

Approximation Hardness of Bounded Degree MIN-CSP and MIN-BISECTION

We consider bounded occurrence (degree) instances of a minimum
constraint satisfaction problem MIN-LIN2 and a MIN-BISECTION problem for
graphs. MIN-LIN2 is an optimization problem for a given system of linear
equations mod 2 to construct a solution that satisfies the minimum number
of them. E3-OCC-MIN-E3-LIN2 ... more >>>


TR13-066 | 25th April 2013
Marek Karpinski, Richard Schmied

Approximation Hardness of Graphic TSP on Cubic Graphs

We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used ... more >>>


TR03-049 | 25th June 2003
Piotr Berman, Marek Karpinski, Alexander D. Scott

Approximation Hardness of Short Symmetric Instances of MAX-3SAT

We prove approximation hardness of short symmetric instances
of MAX-3SAT in which each literal occurs exactly twice, and
each clause is exactly of size 3. We display also an explicit
approximation lower bound for that problem. The bound two on
the number ... more >>>


TR00-089 | 1st December 2000
Lars Engebretsen, Marek Karpinski

Approximation Hardness of TSP with Bounded Metrics

Revisions: 1

The general asymmetric (and metric) TSP is known to be approximable
only to within an O(log n) factor, and is also known to be
approximable within a constant factor as soon as the metric is
bounded. In this paper we study the asymmetric and symmetric TSP
problems with bounded metrics ... more >>>


TR00-058 | 1st August 2000
Martin Sauerhoff

Approximation of Boolean Functions by Combinatorial Rectangles

This paper deals with the number of monochromatic combinatorial
rectangles required to approximate a Boolean function on a constant
fraction of all inputs, where each rectangle may be defined with
respect to its own partition of the input variables. The main result
of the paper is that the number of ... more >>>


TR06-124 | 25th September 2006
Wenceslas Fernandez de la Vega, Ravi Kannan, Marek Karpinski

Approximation of Global MAX-CSP Problems

We study the problem of absolute approximability of MAX-CSP problems with the global constraints. We prove existence of an efficient sampling method for the MAX-CSP class of problems with linear global constraints and bounded feasibility gap. It gives for the first time a polynomial in epsilon^-1 sample complexity bound for ... more >>>


TR12-110 | 4th September 2012
Siu On Chan

Approximation Resistance from Pairwise Independent Subgroups

We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain,
whenever k is larger than the domain size. This follows from our main result concerning predicates
over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that
is ... more >>>


TR12-040 | 17th April 2012
Sangxia Huang

Approximation Resistance on Satisfiable Instances for Predicates Strictly Dominating Parity

In this paper, we study the approximability of Max CSP($P$) where $P$ is a Boolean predicate. We prove that assuming Khot's $d$-to-1 Conjecture, if the set of accepting inputs of $P$ strictly contains all inputs with even (or odd) parity, then it is NP-hard to approximate Max CSP($P$) better than ... more >>>


TR08-009 | 7th December 2007
Per Austrin, Elchanan Mossel

Approximation Resistant Predicates From Pairwise Independence

We study the approximability of predicates on $k$ variables from a
domain $[q]$, and give a new sufficient condition for such predicates
to be approximation resistant under the Unique Games Conjecture.
Specifically, we show that a predicate $P$ is approximation resistant
if there exists a balanced pairwise independent distribution over
more >>>


TR01-065 | 10th August 2001
Chandra Chekuri, Sanjeev Khanna

Approximation Schemes for Preemptive Weighted Flow Time

We present the first approximation schemes for minimizing weighted flow time
on a single machine with preemption. Our first result is an algorithm that
computes a $(1+\eps)$-approximate solution for any instance of weighted flow
time in $O(n^{O(\ln W \ln P/\eps^3)})$ time; here $P$ is the ratio ... more >>>


TR06-074 | 24th April 2006
Shahar Dobzinski, Noam Nisan

Approximations by Computationally-Efficient VCG-Based Mechanisms

We consider computationally-efficient incentive-compatible
mechanisms that use the VCG payment scheme, and study how well they
can approximate the social welfare in auction settings. We obtain a
$2$-approximation for multi-unit auctions, and show that this is
best possible, even though from a purely computational perspective
an FPTAS exists. For combinatorial ... more >>>


TR99-011 | 14th April 1999
Matthias Krause, Petr Savicky, Ingo Wegener

Approximations by OBDDs and the variable ordering problem


Ordered binary decision diagrams (OBDDs) and their variants
are motivated by the need to represent Boolean functions
in applications. Research concerning these applications leads
also to problems and results interesting from theoretical
point of view. In this paper, methods from communication
complexity and ... more >>>


TR09-114 | 13th November 2009
Emanuele Viola

Are all distributions easy?

Complexity theory typically studies the complexity of computing a function $h(x) : \{0,1\}^n \to \{0,1\}^m$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits.

Our main results are:

\begin{itemize}
\item There are explicit $AC^0$ circuits of ... more >>>


TR15-160 | 30th September 2015
Clement Canonne

Are Few Bins Enough: Testing Histogram Distributions

Revisions: 1

A probability distribution over an ordered universe $[n]=\{1,\dots,n\}$ is said to be a $k$-histogram if it can be represented as a piecewise-constant function over at most $k$ contiguous intervals. We study the following question: given samples from an arbitrary distribution $D$ over $[n]$, one must decide whether $D$ is a ... more >>>


TR09-089 | 26th September 2009
Guy Rothblum, Salil Vadhan

Are PCPs Inherent in Efficient Arguments?

Starting with Kilian (STOC `92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collision-resistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC `07) raised the question of whether PCPs ... more >>>


TR15-152 | 16th September 2015
Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

Are Short Proofs Narrow? QBF Resolution is not Simple.

The groundbreaking paper `Short proofs are narrow - resolution made simple' by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in ... more >>>


TR09-057 | 23rd June 2009
Yonatan Bilu, Nathan Linial

Are stable instances easy?

We introduce the notion of a stable instance for a discrete
optimization problem, and argue that in many practical situations
only sufficiently stable instances are of interest. The question
then arises whether stable instances of NP--hard problems are
easier to solve. In particular, whether there exist algorithms
that solve correctly ... more >>>


TR15-145 | 5th September 2015
Eric Allender, Asa Goodwillie

Arithmetic circuit classes over Zm

We continue the study of the complexity classes VP(Zm) and LambdaP(Zm) which was initiated in [AGM15]. We distinguish between “strict” and “lax” versions of these classes and prove some new equalities and inclusions between these arithmetic circuit classes and various subclasses of ACC^1.

more >>>

TR13-028 | 14th February 2013
Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma

Arithmetic Circuit Lower Bounds via MaxRank

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove
super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results :

$\bullet$ As ... more >>>


TR09-026 | 17th February 2009
Vikraman Arvind, Pushkar Joglekar

Arithmetic Circuit Size, Identity Testing, and Finite Automata

Let $\F\{x_1,x_2,\cdots,x_n\}$ be the noncommutative polynomial
ring over a field $\F$, where the $x_i$'s are free noncommuting
formal variables. Given a finite automaton $\A$ with the $x_i$'s as
alphabet, we can define polynomials $\f( mod A)$ and $\f(div A)$
obtained by natural operations that we ... more >>>


TR15-194 | 30th November 2015
Mrinal Kumar, Shubhangi Saraf

Arithmetic circuits with locally low algebraic rank

Revisions: 1

In recent years there has been a flurry of activity proving lower bounds for
homogeneous depth-4 arithmetic circuits [GKKS13, FLMS14, KLSS14, KS14c], which has brought us very close to statements that are known to imply VP $\neq$ VNP. It is a big question to go beyond homogeneity, and in ... more >>>


TR08-048 | 8th April 2008
Meena Mahajan, B. V. Raghavendra Rao

Arithmetic circuits, syntactic multilinearity, and the limitations of skew formulae

Functions in arithmetic NC1 are known to have equivalent constant
width polynomial degree circuits, but the converse containment is
unknown. In a partial answer to this question, we show that syntactic
multilinear circuits of constant width and polynomial degree can be
depth-reduced, though the resulting circuits need not be ... more >>>


TR08-062 | 11th June 2008
Manindra Agrawal, V. Vinay

Arithmetic Circuits: A Chasm at Depth Four

We show that proving exponential lower bounds on depth four arithmetic
circuits imply exponential lower bounds for unrestricted depth arithmetic
circuits. In other words, for exponential sized circuits additional depth
beyond four does not help.

We then show that a complete black-box derandomization of Identity Testing problem for depth four ... more >>>


TR13-026 | 11th February 2013
Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

Arithmetic circuits: A chasm at depth three

Revisions: 1

We show that, over $\mathbb{C}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively by an algebraic branching program of size $s$) then it can also be computed by a depth three circuit (i.e. a $\Sigma \Pi \Sigma$-circuit) of size ... more >>>


TR99-008 | 19th March 1999
Eric Allender, Vikraman Arvind, Meena Mahajan

Arithmetic Complexity, Kleene Closure, and Formal Power Series

Revisions: 1 , Comments: 1

The aim of this paper is to use formal power series techniques to
study the structure of small arithmetic complexity classes such as
GapNC^1 and GapL. More precisely, we apply the Kleene closure of
languages and the formal power series operations of inversion and
root ... more >>>


TR15-061 | 14th April 2015
Benny Applebaum, Jonathan Avron, Christina Brzuska

Arithmetic Cryptography

Revisions: 1

We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a ... more >>>


TR01-095 | 12th November 2001
Hubie Chen

Arithmetic Versions of Constant Depth Circuit Complexity Classes

The boolean circuit complexity classes
AC^0 \subseteq AC^0[m] \subseteq TC^0 \subseteq NC^1 have been studied
intensely. Other than NC^1, they are defined by constant-depth
circuits of polynomial size and unbounded fan-in over some set of
allowed gates. One reason for interest in these classes is that they
contain the ... more >>>


TR07-087 | 11th July 2007
Nutan Limaye, Meena Mahajan, B. V. Raghavendra Rao

Arithmetizing classes around NC^1 and L

The parallel complexity class NC^1 has many equivalent models such as
polynomial size formulae and bounded width branching
programs. Caussinus et al. \cite{CMTV} considered arithmetizations of
two of these classes, #NC^1 and #BWBP. We further this study to
include arithmetization of other classes. In particular, we show that
counting paths ... more >>>


TR09-055 | 10th June 2009
Venkatesan Chakaravarthy, Sambuddha Roy

Arthur and Merlin as Oracles

We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) the Arthur-Merlin class.
Our main results are the following: (i) $BPP^{NP}_{||} \subseteq P^{prAM}_{||}$; (ii) $S_2^p \subseteq P^{prAM}$. In addition to providing new upperbounds for the classes $S_2^p$ and $BPP^{NP}_{||}$, these results are interesting ... more >>>


TR97-054 | 17th November 1997
Ran Raz, Gábor Tardos, Oleg Verbitsky, Nikolay Vereshchagin

Arthur-Merlin Games in Boolean Decision Trees

It is well known that probabilistic boolean decision trees
cannot be much more powerful than deterministic ones (N.~Nisan, SIAM
Journal on Computing, 20(6):999--1007, 1991). Motivated by a question
if randomization can significantly speed up a nondeterministic
computation via a boolean decision tree, we address structural
properties of Arthur-Merlin games ... more >>>


TR13-020 | 2nd February 2013
Tom Gur, Ran Raz

Arthur-Merlin Streaming Complexity

We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical $\mathcal{AM}$ streaming algorithm for a wide class of data stream problems. The algorithm offers a tradeoff between the length of the proof and the space complexity that is needed to verify it.

... more >>>

TR09-001 | 26th November 2008
Venkatesan Guruswami

Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes

Algebraic codes that achieve list decoding capacity were recently
constructed by a careful ``folding'' of the Reed-Solomon code. The
``low-degree'' nature of this folding operation was crucial to the list
decoding algorithm. We show how such folding schemes conducive to list
decoding arise out of the Artin-Frobenius automorphism at primes ... more >>>


TR13-105 | 29th July 2013
Raghu Meka, Avi Wigderson

Association schemes, non-commutative polynomial concentration, and sum-of-squares lower bounds for planted clique

Revisions: 1

Finding cliques in random graphs and the closely related ``planted'' clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = ... more >>>


TR03-038 | 15th May 2003
Julia Chuzhoy, Sudipto Guha, Sanjeev Khanna, Seffi Naor

Asymmetric k-center is log^*n-hard to Approximate

We show that the asymmetric $k$-center problem is
$\Omega(\log^* n)$-hard to approximate unless
${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$.
Since an $O(\log^* n)$-approximation algorithm is known
for this problem, this essentially resolves the approximability
of this problem. This is the first natural problem
whose approximability threshold does not polynomially ... more >>>


TR99-048 | 7th December 1999
Beate Bollig, Ingo Wegener

Asymptotically Optimal Bounds for OBDDs and the Solution of Some Basic OBDD Problems

Ordered binary decision diagrams (OBDDs) are nowadays the
most common dynamic data structure or representation type
for Boolean functions. Among the many areas of application
are verification, model checking, and computer aided design.
For many functions it is easy to estimate the OBDD ... more >>>


TR95-026 | 7th June 1995
Claus-Peter Schnorr, Horst Helmut Hoerner

Attacking the Chor-Rivest Cryptosystem by Improved Lattice Reduction

We introduce new algorithms for lattice basis reduction that are
improvements of the LLL-algorithm. We demonstrate the power of
these algorithms by solving random subset sum problems of
arbitrary density with 74 and 82 many weights, by breaking the
Chor-Rivest cryptoscheme in dimensions 103 and 151 ... more >>>


TR98-076 | 13th November 1998
Nader H. Bshouty, Jeffrey J. Jackson, Christino Tamon

Attribute Efficient PAC Learning of DNF with Membership Queries under the Uniform Distribution

We study attribute efficient learning in the PAC learning model with
membership queries. First, we give an {\it attribute efficient}
PAC-learning algorithm for DNF with membership queries under the
uniform distribution. Previous algorithms for DNF have sample size
polynomial in the number of attributes $n$. Our algorithm is the
first ... more >>>


TR12-056 | 1st May 2012
Rocco Servedio, Li-Yang Tan, Justin Thaler

Attribute-Efficient Learning and Weight-Degree Tradeoffs for Polynomial Threshold Functions

Revisions: 1

We study the challenging problem of learning decision lists attribute-efficiently, giving both positive and negative results.

Our main positive result is a new tradeoff between the running time and mistake bound for learning length-$k$ decision lists over $n$ Boolean variables. When the allowed running time is relatively high, our new ... more >>>


TR13-188 | 13th December 2013
Christian Glaßer, Maximilian Witek

Autoreducibility and Mitoticity of Logspace-Complete Sets for NP and Other Classes

We study the autoreducibility and mitoticity of complete sets for NP and other complexity classes, where the main focus is on logspace reducibilities. In particular, we obtain:
- For NP and all other classes of the PH: each logspace many-one-complete set is logspace Turing-autoreducible.
- For P, the delta-levels of ... more >>>


TR13-047 | 27th March 2013
Christian Glaßer, Dung Nguyen, Christian Reitwießner, Alan Selman, Maximilian Witek

Autoreducibility of Complete Sets for Log-Space and Polynomial-Time Reductions

Comments: 1

We investigate the autoreducibility and mitoticity of complete sets for several classes with respect to different polynomial-time and logarithmic-space reducibility notions.

Previous work in this area focused on polynomial-time reducibility notions. Here we obtain new mitoticity and autoreducibility results for the classes EXP and NEXP with respect to some restricted ... more >>>


TR16-012 | 21st January 2016
John Hitchcock, Hadi Shafei

Autoreducibility of NP-Complete Sets

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following:

- For every $k \geq 2$, there is a $k$-T-complete set for NP that is $k$-T autoreducible, but is not $k$-tt autoreducible ... more >>>


TR05-011 | 21st December 2004
Christian Glaßer, Mitsunori Ogihara, A. Pavan, Alan L. Selman, Liyu Zhang

Autoreducibility, Mitoticity, and Immunity

We show the following results regarding complete sets:

NP-complete sets and PSPACE-complete sets are many-one
autoreducible.

Complete sets of any level of PH, MODPH, or
the Boolean hierarchy over NP are many-one autoreducible.

EXP-complete sets are many-one mitotic.

NEXP-complete sets are weakly many-one mitotic.

PSPACE-complete sets are weakly Turing-mitotic.

... more >>>

TR13-054 | 4th April 2013
Yuval Filmus, Toniann Pitassi, Robert Robere, Stephen A. Cook

Average Case Lower Bounds for Monotone Switching Networks

Revisions: 1

An approximate computation of a Boolean function by a circuit or switching network is a computation which computes the function correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas ... more >>>


TR95-019 | 14th April 1995
Jin-Yi Cai, Alan L. Selman

Average time complexity classes


TR06-073 | 8th June 2006
Andrej Bogdanov, Luca Trevisan

Average-Case Complexity

Revisions: 1

We survey the theory of average-case complexity, with a
focus on problems in NP.

more >>>

TR03-031 | 8th April 2003
Birgit Schelm

Average-Case Complexity Theory of Approximation Problems

Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>>


TR17-039 | 28th February 2017
Marshall Ball, Alon Rosen, Manuel Sabin, Prashant Nalini Vasudevan

Average-Case Fine-Grained Hardness

We present functions that can be computed in some fixed polynomial time but are hard on average for any algorithm that runs in slightly smaller time, assuming widely-conjectured worst-case hardness for problems from the study of fine-grained complexity. Unconditional constructions of such functions are known from before (Goldmann et al., ... more >>>


TR98-037 | 29th June 1998
Johannes Köbler, Rainer Schuler

Average-Case Intractability vs. Worst-Case Intractability

We use the assumption that all sets in NP (or other levels
of the polynomial-time hierarchy) have efficient average-case
algorithms to derive collapse consequences for MA, AM, and various
subclasses of P/poly. As a further consequence we show for
C in {P(PP), PSPACE} that ... more >>>


TR15-191 | 26th November 2015
Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan

Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is \epsilon_d > 0 such that Parity has correlation at most 1/n^{\Omega(1)} with depth-d threshold circuits which have at most
n^{1+\epsilon_d} ... more >>>


TR12-062 | 17th May 2012
Ilan Komargodski, Ran Raz

Average-Case Lower Bounds for Formula Size

Revisions: 2

We give an explicit function $h:\{0,1\}^n\to\{0,1\}$ such that any deMorgan formula of size $O(n^{2.499})$ agrees with $h$ on at most $\frac{1}{2} + \epsilon$ fraction of the inputs, where $\epsilon$ is exponentially small (i.e. $\epsilon = 2^{-n^{\Omega(1)}}$). Previous lower bounds for formula size were obtained for exact computation.

The same ... more >>>


TR11-006 | 20th January 2011
Sebastian Müller, Iddo Tzameret

Average-Case Separation in Proof Complexity: Short Propositional Refutations for Random 3CNF Formulas

Revisions: 1

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>>


TR10-055 | 31st March 2010
Eric Allender

Avoiding Simplicity is Complex

Revisions: 2

It is a trivial observation that every decidable set has strings of length $n$ with Kolmogorov complexity $\log n + O(1)$ if it has any strings of length $n$ at all. Things become much more interesting when one asks whether a similar property holds when one
considers *resource-bounded* Kolmogorov complexity. ... more >>>




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