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TR05-147 | 5th December 2005
Christian Glaßer, Stephen Travers

Machines that can Output Empty Words

We propose the e-model for leaf languages which generalizes the known balanced and unbalanced concepts. Inspired by the neutral behavior of rejecting paths of NP machines, we allow transducers to output empty words.

The paper explains several advantages of the new model. A central aspect is that it allows us ... more >>>

TR17-003 | 24th November 2016
Yi Deng

Magic Adversaries Versus Individual Reduction: Science Wins Either Way

Revisions: 1

We prove that \emph{at least} one of the following statements is true:

-- (Infinitely-often) Public-key encryption and key agreement can be based on injective one-way functions;
-- For every inverse polynomial $\epsilon$, the 4-round protocol from [Feige and Shamir, STOC 90] is distributional concurrent zero knowledge for any ... more >>>

TR14-159 | 27th November 2014
A. C. Cem Say, Abuzer Yakaryilmaz

Magic coins are useful for small-space quantum machines

Although polynomial-time probabilistic Turing machines can utilize uncomputable transition probabilities to recognize uncountably many languages with bounded error when allowed to use logarithmic space, it is known that such ``magic coins'' give no additional computational power to constant-space versions of those machines. We show that adding a few quantum bits ... more >>>

TR14-173 | 13th December 2014
Igor Carboni Oliveira, Rahul Santhanam

Majority is incompressible by AC$^0[p]$ circuits

Revisions: 1

We consider $\cal C$-compression games, a hybrid model between computational and communication complexity. A $\cal C$-compression game for a function $f \colon \{0,1\}^n \to \{0,1\}$ is a two-party communication game, where the first party Alice knows the entire input $x$ but is restricted to use strategies computed by $\cal C$-circuits, ... more >>>

TR06-003 | 8th January 2006
Joshua Buresh-Oppenheim, Rahul Santhanam

Making Hard Problems Harder

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as ... more >>>

TR97-014 | 25th April 1997
Klaus Reinhardt, Eric Allender

Making Nondeterminism Unambiguous

We show that in the context of nonuniform complexity,
nondeterministic logarithmic space bounded computation can be made
unambiguous. An analogous result holds for the class of problems
reducible to context-free languages. In terms of complexity classes,
this can be stated as:
NL/poly = UL/poly
LogCFL/poly ... more >>>

TR12-037 | 10th April 2012
Alexander A. Sherstov

Making Polynomials Robust to Noise

A basic question in any computational model is how to reliably compute a given function when the inputs or intermediate computations are subject to noise at a constant rate. Ideally, one would like to use at most a constant factor more resources compared to the noise-free case. This question has ... more >>>

TR10-104 | 29th June 2010
Paul Beame, Widad Machmouchi

Making RAMs Oblivious Requires Superlogarithmic Overhead

Revisions: 3 , Comments: 1

We prove a time-space tradeoff lower bound of $T =
\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ for
randomized oblivious branching programs to compute $1GAP$, also
known as the pointer jumping problem, a problem for which there is a
simple deterministic time $n$ and space $O(\log n)$ RAM (random
access machine) algorithm.

In ... more >>>

TR11-142 | 2nd November 2011
Boaz Barak, Parikshit Gopalan, Johan Hastad, Raghu Meka, Prasad Raghavendra, David Steurer

Making the long code shorter, with applications to the Unique Games Conjecture

Revisions: 1

The long code is a central tool in hardness of approximation, especially in
questions related to the unique games conjecture. We construct a new code that
is exponentially more ecient, but can still be used in many of these applications.
Using the new code we obtain exponential improvements over several ... more >>>

TR16-052 | 7th April 2016
Gil Cohen

Making the Most of Advice: New Correlation Breakers and Their Applications

A typical obstacle one faces when constructing pseudorandom objects is undesired correlations between random variables. Identifying this obstacle and constructing certain types of "correlation breakers" was central for recent exciting advances in the construction of multi-source and non-malleable extractors. One instantiation of correlation breakers is correlation breakers with advice. These ... more >>>

TR02-034 | 18th April 2002
Andrei Bulatov

Mal'tsev constraints are tractable

A wide variety of combinatorial problems can be represented in the form of the Constraint Satisfaction Problem (CSP). The general CSR is known to be NP-complete, however, some restrictions on the possible form of constraints may lead to a tractable subclass. Jeavons and coauthors have shown that the complexity of ... more >>>

TR04-097 | 2nd November 2004
Víctor Dalmau

Malt'sev Constraints made Simple

We give in this paper a different and simpler proof of the tractability of Mal'tsev contraints.

more >>>

TR10-023 | 23rd February 2010
Adam Klivans, Homin Lee, Andrew Wan

Mansour’s Conjecture is True for Random DNF Formulas

Revisions: 3

In 1994, Y. Mansour conjectured that for every DNF formula on $n$ variables with $t$ terms there exists a polynomial $p$ with $t^{O(\log (1/\epsilon))}$ non-zero coefficients such that $\E_{x \in \{0,1\}}[(p(x)-f(x))^2] \leq \epsilon$. We make the first progress on this conjecture and show that it is true for several natural ... more >>>

TR11-133 | 4th October 2011
Maurice Jansen, Rahul Santhanam

Marginal Hitting Sets Imply Super-Polynomial Lower Bounds for Permanent

Suppose $f$ is a univariate polynomial of degree $r=r(n)$ that is computed by a size $n$ arithmetic circuit.
It is a basic fact of algebra that a nonzero univariate polynomial of degree $r$ can vanish on at most $r$ points. This implies that for checking whether $f$ is identically zero, ... more >>>

TR05-045 | 12th April 2005
Philippe Moser

Martingale Families and Dimension in P

Revisions: 1

We introduce a new measure notion on small complexity classes (called F-measure), based on martingale families,
that get rid of some drawbacks of previous measure notions:
martingale families can make money on all strings,
and yield random sequences with an equal frequency of 0's and 1's.
As applications to F-measure,
more >>>

TR12-107 | 30th August 2012
Brendan Juba, Ryan Williams

Massive Online Teaching to Bounded Learners

We consider a model of teaching in which the learners are consistent and have bounded state, but are otherwise arbitrary. The teacher is non-interactive and ``massively open'': the teacher broadcasts a sequence of examples of an arbitrary target concept, intended for every possible on-line learning algorithm to learn from. We ... more >>>

TR13-048 | 27th March 2013
Jin-Yi Cai, Aaron Gorenstein

Matchgates Revisited

We study a collection of concepts and theorems that laid the foundation of matchgate computation. This includes the signature theory of planar matchgates, and the parallel theory of characters of not necessarily planar matchgates. Our aim is to present a unified and, whenever possible, simplified account of this challenging theory. ... more >>>

TR10-012 | 27th January 2010
Zeev Dvir, Parikshit Gopalan, Sergey Yekhanin

Matching Vector Codes

Revisions: 1

An $(r,\delta,\epsilon)$-locally decodable code encodes a $k$-bit message $x$ to an $N$-bit codeword $C(x),$ such that for every $i\in [k],$ the $i$-th message bit can be recovered with probability $1-\epsilon,$ by a randomized decoding procedure that reads only $r$ bits, even if the codeword $C(x)$ is corrupted in up to ... more >>>

TR13-061 | 17th April 2013
Zeev Dvir, Guangda Hu

Matching-Vector Families and LDCs Over Large Modulo

We prove new upper bounds on the size of families of vectors in $\Z_m^n$ with restricted modular inner products, when $m$ is a large integer. More formally, if $\vec{u}_1,\ldots,\vec{u}_t \in \Z_m^n$ and $\vec{v}_1,\ldots,\vec{v}_t \in \Z_m^n$ satisfy $\langle\vec{u}_i,\vec{v}_i\rangle\equiv0\pmod m$ and $\langle\vec{u}_i,\vec{v}_j\rangle\not\equiv0\pmod m$ for all $i\neq j\in[t]$, we prove that $t \leq ... more >>>

TR15-079 | 7th May 2015
Oded Goldreich, Avishay Tal

Matrix Rigidity of Random Toeplitz Matrices

We prove that random $n$-by-$n$ Toeplitz (alternatively Hankel) matrices over $GF(2)$ have rigidity $\Omega(\frac{n^3}{r^2\log n})$ for rank $r \ge \sqrt{n}$, with high probability. This improves, for $r = o(n/\log n \log\log n)$, over the $\Omega(\frac{n^2}{r} \cdot\log(\frac{n}{r}))$ bound that is known for many explicit matrices.

Our result implies that the explicit ... more >>>

TR97-039 | 11th September 1997
Pierluigi Crescenzi, Luca Trevisan

MAX NP-Completeness Made Easy

We introduce a simple technique to obtain reductions
between optimization constraint satisfaction problems. The
technique uses the probabilistic method to reduce the size of
disjunctions. As a first application, we prove the
MAX NP-completeness of MAX 3SAT without using the PCP theorem
(thus solving an open ... more >>>

TR11-099 | 11th July 2011
Anant Jindal, Gazal Kochar, Manjish Pal

Maximum Matchings via Glauber Dynamics

Revisions: 1 , Comments: 1

In this paper we study the classic problem of computing a maximum cardinality matching in general graphs $G = (V, E)$. This problem has been studied extensively more than four decades. The best known algorithm for this problem till date runs in $O(m \sqrt{n})$ time due to Micali and Vazirani ... more >>>

TR04-040 | 4th May 2004
Venkatesan Guruswami, Alexander Vardy

Maximum-likelihood decoding of Reed-Solomon codes is NP-hard

Maximum-likelihood decoding is one of the central algorithmic
problems in coding theory. It has been known for over 25 years
that maximum-likelihood decoding of general linear codes is
NP-hard. Nevertheless, it was so far unknown whether maximum-
likelihood decoding remains hard for any specific family of
more >>>

TR10-197 | 14th December 2010
Albert Atserias, Elitza Maneva

Mean-payoff games and propositional proofs

We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in $\Sigma_2$-Frege (i.e.~DNF-resolution). This reduces mean-payoff ... more >>>

TR14-057 | 17th April 2014
Manindra Agrawal, Diptarka Chakraborty, Debarati Das, Satyadev Nandakumar

Measure of Non-pseudorandomness and Deterministic Extraction of Pseudorandomness

Revisions: 3

In this paper, we propose a quantification of distributions on a set
of strings, in terms of how close to pseudorandom the distribution
is. The quantification is an adaptation of the theory of dimension of
sets of infinite sequences first introduced by Lutz
We show that this definition ... more >>>

TR02-065 | 26th November 2002
Olivier Powell

Measure on P revisited

We revisit the problem of generalising Lutz's resource bounded measure
(rbm) to small complexity classes.
We propose a definition of a perfect rbm on P,
and give sufficient and necessary conditions for such a measure to exist.
We also revisit $\mu_\tau$, an rbm for P
defined in previous articles (c.f. ... more >>>

TR95-028 | 9th June 1995
Eric Allender, Martin Strauss

Measure on P: Robustness of the Notion

In (Allender and Strauss, FOCS '95), we defined a notion of
measure on the complexity class P (in the spirit of the work of (Lutz,
JCSS '92) that provides a notion of measure on complexity classes at least
as large as E, and the work of (Mayordomo, Phd. ... more >>>

TR94-004 | 12th December 1994
Eric Allender, Martin Strauss

Measure on Small Complexity Classes, with Applications for BPP

We present a notion of resource-bounded measure for P and other
subexponential-time classes. This generalization is based on Lutz's
notion of measure, but overcomes the limitations that cause Lutz's
definitions to apply only to classes at least as large as E. We
present many of the basic properties of this ... more >>>

TR00-076 | 24th August 2000
Juraj Hromkovic, Juhani Karhumaki, Hartmut Klauck, Georg Schnitger, Sebastian Seibert

Measures of Nondeterminism in Finite Automata

While deterministic finite automata seem to be well understood, surprisingly
many important problems
concerning nondeterministic finite automata (nfa's) remain open.

One such problem area is the study of different measures of nondeterminism in
finite automata and the
estimation of the sizes of minimal nondeterministic finite automata. In this
paper the ... more >>>

TR05-004 | 3rd January 2005
Leslie G. Valiant

Memorization and Association on a Realistic Neural Model

A central open question of computational neuroscience is to identify the data structures and algorithms that are used in mammalian cortex to support successive acts of the basic cognitive tasks of memorization and association. This paper addresses the simultaneous challenges of realizing these two distinct tasks with the same data ... more >>>

TR15-126 | 27th July 2015
Jacob Steinhardt, Gregory Valiant, Stefan Wager

Memory, Communication, and Statistical Queries

Revisions: 1

If a concept class can be represented with a certain amount of memory, can it be efficiently learned with the same amount of memory? What concepts can be efficiently learned by algorithms that extract only a few bits of information from each example? We introduce a formal framework for studying ... more >>>

TR11-092 | 2nd June 2011
Doerr Benjamin, Winzen Carola

Memory-Restricted Black-Box Complexity

We show that the black-box complexity with memory restriction one of the $n$-dimensional $\onemax$ function class is at most $2n$. This disproves the $\Theta(n \log n)$ conjecture of Droste, Jansen, and Wegener (Theory of Computing Systems 39 (2006) 525--544).

more >>>

TR05-151 | 7th December 2005
Magnus Bordewich, Martin Dyer, Marek Karpinski

Metric Construction, Stopping Times and Path Coupling.

In this paper we examine the importance of the choice of metric in path coupling, and the relationship of this to \emph{stopping time analysis}. We give strong evidence that stopping time analysis is no more powerful than standard path coupling. In particular, we prove a stronger theorem for path coupling ... more >>>

TR16-128 | 13th August 2016
Irit Dinur

Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover

We show that if gap-3SAT has no sub-exponential time algorithms then a weak form of the sliding scale conjecture holds. Namely, for every $\alpha>0$ any algorithm for $n^\alpha$-approximating the value of label cover must run in time at least $n^{\Omega(\exp(1/\alpha))}$, where $n$ is the size of the instance.

Put differently, ... more >>>

TR09-008 | 15th January 2009
Stasys Jukna, Georg Schnitger

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained
by setting all *-entries to constants 0 or 1. A system of semi-linear
equations over GF(2) has the form Mx=f(x), where M is a completion of
A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate ... more >>>

TR03-002 | 13th December 2002
Stefan Szeider

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Revisions: 1

The deficiency of a propositional formula F in CNF with n variables
and m clauses is defined as m-n. It is known that minimal
unsatisfiable formulas (unsatisfiable formulas which become
satisfiable by removing any clause) have positive deficiency.
Recognition of minimal unsatisfiable formulas is NP-hard, and it was
shown recently ... more >>>

TR02-054 | 5th September 2002
Detlef Sieling

Minimization of Decision Trees is Hard to Approximate

Decision trees are representations of discrete functions with widespread applications in, e.g., complexity theory and data mining and exploration. In these areas it is important to obtain decision trees of small size. The minimization problem for decision trees is known to be NP-hard. In this paper the problem is shown ... more >>>

TR05-126 | 5th November 2005
Eric Allender, Lisa Hellerstein, Paul McCabe, Michael Saks

Minimizing DNF Formulas and AC0 Circuits Given a Truth Table

For circuit classes R, the fundamental computational problem, Min-R,
asks for the minimum R-size of a boolean function presented as a truth
table. Prominent examples of this problem include Min-DNF, and
Min-Circuit (also called MCSP). We begin by presenting a new reduction
proving that Min-DNF is NP-complete. It is significantly ... more >>>

TR15-045 | 1st April 2015
Benny Applebaum, Yuval Ishai, Eyal Kushilevitz

Minimizing Locality of One-Way Functions via Semi-Private Randomized Encodings

Revisions: 1

A one-way function is $d$-local if each of its outputs depends on at most $d$ input bits. In (Applebaum, Ishai, and Kushilevitz, FOCS 2004) it was shown that, under relatively mild assumptions, there exist $4$-local one-way functions (OWFs). This result is not far from optimal as it is not hard ... more >>>

TR17-158 | 23rd October 2017
Eric Allender, Joshua Grochow, Dieter van Melkebeek, Cris Moore, Andrew Morgan

Minimum Circuit Size, Graph Isomorphism, and Related Problems

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions ... more >>>

TR13-057 | 5th April 2013
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, David Zuckerman

Mining Circuit Lower Bound Proofs for Meta-Algorithms

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for ``easy'' Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an $n$-variate Boolean function $f$ computable by some unknown small circuit ... more >>>

TR17-017 | 5th February 2017
Michal Moshkovitz, Dana Moshkovitz

Mixing Implies Lower Bounds for Space Bounded Learning

One can learn any hypothesis class $H$ with $O(\log|H|)$ labeled examples. Alas, learning with so few examples requires saving the examples in memory, and this requires $|X|^{O(\log|H|)}$ memory states, where $X$ is the set of all labeled examples. A question that arises is how many labeled examples are needed in ... more >>>

TR17-116 | 5th July 2017
Michal Moshkovitz, Dana Moshkovitz

Mixing Implies Strong Lower Bounds for Space Bounded Learning

With any hypothesis class one can associate a bipartite graph whose vertices are the hypotheses H on one side and all possible labeled examples X on the other side, and an hypothesis is connected to all the labeled examples that are consistent with it. We call this graph the hypotheses ... more >>>

TR09-111 | 5th November 2009
Walid Gomaa

Model-Theoretic Characterization of Complexity Classes

Model theory is a branch of mathematical logic that investigates the
logical properties of mathematical structures. It has been quite
successfully applied to computational complexity resulting in an
area of research called descriptive complexity theory. Descriptive
complexity is essentially a syntactical characterization of
complexity classes using logical formalisms. However, there ... more >>>

TR05-120 | 14th October 2005
Sashka Davis, Russell Impagliazzo

Models of Greedy Algorithms for Graph Problems

Borodin, Nielsen, and Rackoff gave a model of greedy-like algorithms for scheduling problems and Angelopoulos and Borodin extended their work to facility location and set cover problems. We generalize their notion to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an ... more >>>

TR96-001 | 10th January 1996
Manindra Agrawal, Richard Beigel, Thomas Thierauf

Modulo Information from Nonadaptive Queries to NP

The classes of languages accepted by nondeterministic polynomial-time
Turing machines (NP machines, in short) that have restricted access to
an NP oracle --- the machines can ask k queries to the NP oracle and
the answer they receive is the number of queries ... more >>>

TR13-008 | 7th January 2013
Adam Klivans, Raghu Meka

Moment-Matching Polynomials

We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem and deviate significantly from known Fourier-based methods, which require the underlying distribution to have some product ... more >>>

TR17-175 | 13th November 2017
Ankit Garg, Mika Göös, Pritish Kamath, Dmitry Sokolov

Monotone Circuit Lower Bounds from Resolution

For any unsatisfiable CNF formula $F$ that is hard to refute in the Resolution proof system, we show that a gadget-composed version of $F$ is hard to refute in any proof system whose lines are computed by efficient communication protocols---or, equivalently, that a monotone function associated with $F$ has large ... more >>>

TR11-121 | 12th September 2011
Oded Goldreich, Rani Izsak

Monotone Circuits: One-Way Functions versus Pseudorandom Generators

We study the computability of one-way functions and pseudorandom generators
by monotone circuits, showing a substantial gap between the two:
On one hand, there exist one-way functions that are computable
by (uniform) polynomial-size monotone functions, provided (of course)
that one-way functions exist at all.
On the other hand, ... more >>>

TR02-007 | 14th January 2002
Pavel Pudlak

Monotone complexity and the rank of matrices

Comments: 1

The rank of a matrix has been used a number of times to prove lower
bounds on various types of complexity. In particular it has been used
for the size of monotone formulas and monotone span programs. In most
cases that this approach was used, there is not a single ... more >>>

TR09-135 | 10th December 2009
Zeev Dvir, Avi Wigderson

Monotone expanders - constructions and applications

The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to:
(1) Constant degree dimension expanders in finite ... more >>>

TR15-171 | 28th October 2015
Joshua Grochow

Monotone projection lower bounds from extended formulation lower bounds

Revisions: 2 , Comments: 1

In this short note, we show that the permanent is not complete for non-negative polynomials in $VNP_{\mathbb{R}}$ under monotone p-projections. In particular, we show that Hamilton Cycle polynomial and the cut polynomials are not monotone p-projections of the permanent. To prove this we introduce a new connection between monotone projections ... more >>>

TR00-008 | 20th January 2000
Albert Atserias, Nicola Galesi, Ricard Gavaldà

Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole
Principle (PHP) in a monotone variant of the Gentzen Calculus, also
known as Geometric Logic. We show that the standard encoding
of the PHP as a monotone sequent admits quasipolynomial-size proofs
in this system. This result is a consequence of ... more >>>

TR11-025 | 19th February 2011
Yang Li

Monotone Rank and Separations in Computational Complexity

Revisions: 1 , Comments: 1

In the paper, we introduce the concept of monotone rank, and using it as a powerful tool, we obtain several important and strong separation results in computational complexity.


\item We show a super-exponential separation between monotone and non-monotone computation in the non-commutative model, and thus give the answer to ... more >>>

TR00-087 | 14th November 2000
Albert Atserias, Nicola Galesi, Pavel Pudlak

Monotone simulations of nonmonotone propositional proofs

We show that an LK proof of size $m$ of a monotone sequent (a sequent

that contains only formulas in the basis $\wedge,\vee$) can be turned

into a proof containing only monotone formulas of size $m^{O(\log m)}$

and with the number of proof lines polynomial in $m$. Also we show

... more >>>

TR10-048 | 24th March 2010
David García Soriano, Arie Matsliah, Sourav Chakraborty, Jop Briet

Monotonicity Testing and Shortest-Path Routing on the Cube

We study the problem of monotonicity testing over the hypercube. As
previously observed in several works, a positive answer to a natural question about routing
properties of the hypercube network would imply the existence of efficient
monotonicity testers. In particular, if any $\ell$ disjoint source-sink pairs
on the directed hypercube ... more >>>

TR17-167 | 3rd November 2017
Chin Ho Lee, Emanuele Viola

More on bounded independence plus noise: Pseudorandom generators for read-once polynomials

We construct pseudorandom generators with improved seed length for
several classes of tests. First we consider the class of read-once
polynomials over GF(2) in $m$ variables. For error $\e$ we obtain seed
length $\tilde O (\log(m/\e)) \log(1/\e)$, where $\tilde O$ hides lower-order
terms. This is optimal up to the factor ... more >>>

TR17-097 | 31st May 2017
Itay Berman, Akshay Degwekar, Ron Rothblum, Prashant Nalini Vasudevan

Multi Collision Resistant Hash Functions and their Applications

Revisions: 1

Collision resistant hash functions are functions that shrink their input, but for which it is computationally infeasible to find a collision, namely two strings that hash to the same value (although collisions are abundant).

In this work we study multi-collision resistant hash functions (MCRH) a natural relaxation of collision resistant ... more >>>

TR15-015 | 30th January 2015
Neeraj Kayal, Chandan Saha

Multi-$k$-ic depth three circuit lower bound

In a multi-$k$-ic depth three circuit every variable appears in at most $k$ of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables ... more >>>

TR17-099 | 5th June 2017
Nir Bitansky, Omer Paneth, Yael Tauman Kalai

Multi-Collision Resistance: A Paradigm for Keyless Hash Functions

Revisions: 1

We study multi-collision-resistant hash functions --- a natural relaxation of collision-resistant hashing that only guarantees the intractability of finding many (rather than two) inputs that map to the same image. An appealing feature of such hash functions is that unlike their collision-resistant counterparts, they do not necessarily require a key. ... more >>>

TR00-015 | 16th February 2000
Andrej Muchnik, Alexej Semenov

Multi-conditional Descriptions and Codes in Kolmogorov Complexity

TR03-067 | 14th August 2003
Ran Raz

Multi-Linear Formulas for Permanent and Determinant are of Super-Polynomial Size

An arithmetic formula is multi-linear if the polynomial computed
by each of its sub-formulas is multi-linear. We prove that any
multi-linear arithmetic formula for the permanent or the
determinant of an $n \times n$ matrix is of size super-polynomial
in $n$.

more >>>

TR14-149 | 10th November 2014
Kai-Min Chung, Xin Li, Xiaodi Wu

Multi-Source Randomness Extractors Against Quantum Side Information, and their Applications

We study the problem of constructing multi-source extractors in the quantum setting, which extract almost uniform random bits against quantum side information collected from several initially independent classical random sources. This is a natural generalization of seeded randomness extraction against quantum side information and classical independent source extraction. With new ... more >>>

TR13-011 | 10th January 2013
Nader Bshouty

Multilinear Complexity is Equivalent to Optimal Tester Size

In this paper we first show that Tester for an $F$-algebra $A$
and multilinear forms (see Testers and their Applications ECCC 2012) is equivalent to multilinear
algorithm for the product of elements in $A$
(see Algebraic
complexity theory. vol. 315, Springer-Verlag). Our
result is constructive in deterministic polynomial time. ... more >>>

TR03-079 | 8th November 2003
Scott Aaronson

Multilinear Formulas and Skepticism of Quantum Computing

Several researchers, including Leonid Levin, Gerard 't Hooft, and
Stephen Wolfram, have argued that quantum mechanics will break down
before the factoring of large numbers becomes possible. If this is
true, then there should be a natural "Sure/Shor separator" -- that is,
a set of quantum ... more >>>

TR07-085 | 2nd September 2007
Ran Raz, Amir Yehudayoff

Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors

We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

1. Noise-resistant. A syntactically multilinear ... more >>>

TR04-042 | 21st May 2004
Ran Raz

Multilinear-$NC_1$ $\ne$ Multilinear-$NC_2$

An arithmetic circuit or formula is multilinear if the polynomial
computed at each of its wires is multilinear.
We give an explicit example for a polynomial $f(x_1,...,x_n)$,
with coefficients in $\{0,1\}$, such that over any field:
1) $f$ can be computed by a polynomial-size multilinear circuit
of depth $O(\log^2 ... more >>>

TR08-082 | 11th September 2008
Paul Beame, Trinh Huynh

Multiparty Communication Complexity and Threshold Circuit Size of AC^0

Revisions: 2

We prove an n^{Omega(1)}/2^{O(k)} lower bound on the randomized k-party communication complexity of read-once depth 4 AC^0 functions in the number-on-forehead (NOF) model for O(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC^0 function for omega(log log n) players. For ... more >>>

TR08-061 | 11th June 2008
Paul Beame, Trinh Huynh

Multiparty Communication Complexity of AC^0

Revisions: 1

We prove n^Omega(1) lower bounds on the multiparty communication complexity of AC^0 functions in the number-on-forehead (NOF) model for up to Theta(log n) players. These are the first lower bounds for any AC^0 function for omega(loglog n) players. In particular we show that there are families of depth 3 read-once ... more >>>

TR08-002 | 19th December 2007
Arkadev Chattopadhyay, Anil Ada

Multiparty Communication Complexity of Disjointness

Revisions: 3

We extend the 'Generalized Discrepancy' technique suggested by Sherstov to the `Number on the Forehead' model of multiparty communication. This allows us to prove strong lower bounds of n^{\Omega(1)} on the communication needed by k players to compute the Disjointness function, provided $k$ is a constant. In general, our method ... more >>>

TR16-160 | 26th October 2016
Irit Dinur, Prahladh Harsha, Rakesh Venkat, Henry Yuen

Multiplayer parallel repetition for expander games

Revisions: 1

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially ... more >>>

TR95-017 | 27th March 1995
Claudia Bertram, Thomas Hofmeister

Multiple Product Modulo Arbitrary Numbers

We consider the threshold circuit complexity of computing
the multiple product modulo m in threshold circuits.

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TR16-185 | 18th November 2016
Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

Multiplicative Approximations for Polynomial Optimization Over the Unit Sphere

Revisions: 1

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in many diverse contexts ranging from tensor and operator norms to graph expansion to quantum information ... more >>>

TR06-006 | 16th December 2005
Alexander Shen

Multisource algorithmic information theory

Multisource information theory in Shannon setting is well known. In this article we try to develop its algorithmic information theory counterpart and use it as the general framework for many interesting questions about Kolmogorov complexity.

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TR08-096 | 8th September 2008
Andrew Drucker

Multitask Efficiencies in the Decision Tree Model

In Direct Sum problems [KRW], one tries to show that for a given computational model, the complexity of computing a collection $F = \{f_1(x_1), \ldots f_l(x_l)\}$ of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ... more >>>

TR10-202 | 9th December 2010
Bin Fu

Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable unless P=NP

We investigate the complexity of integration and
derivative for multivariate polynomials in the standard computation
model. The integration is in the unit cube $[0,1]^d$ for a
multivariate polynomial, which has format
x_d)=p_1(x_1,\cdots, x_d)p_2(x_1,\cdots, x_d)\cdots p_k(x_1,\cdots,
where each $p_i(x_1,\cdots, x_d)=\sum_{j=1}^d q_j(x_j)$ with
all single variable polynomials $q_j(x_j)$ ... more >>>

TR14-133 | 15th October 2014
Adam Case, Jack H. Lutz

Mutual Dimension

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory ... more >>>

TR94-021 | 12th December 1994
Lance Fortnow

My Favorite Ten Complexity Theorems of the Past Decade

We review the past ten years in computational complexity theory by
focusing on ten theorems that the author enjoyed the most. We use
each of the theorems as a springboard to discuss work done in
various areas of complexity theory.

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