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TR17-033 | 19th February 2017
Daniel Kane, Shachar Lovett, Sankeerth Rao Karingula

#### Labeling the complete bipartite graph with no zero cycles

Revisions: 2

Assume that the edges of the complete bipartite graph $K_{n,n}$ are labeled with elements of $\mathbb{F}_2^d$, such that the sum over
any simple cycle is nonzero. What is the smallest possible value of $d$? This problem was raised by Gopalan et al. [SODA 2017] as it characterizes the alphabet size ... more >>>

TR04-002 | 8th January 2004
Troy Lee, Dieter van Melkebeek, Harry Buhrman

#### Language Compression and Pseudorandom Generators

The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information ... more >>>

TR05-152 | 9th December 2005
Oded Lachish, Ilan Newman

#### Languages that are Recognized by Simple Counter Automata are not necessarily Testable

Combinatorial property testing deals with the following relaxation of
decision problems: Given a fixed property and an input $f$, one wants
to decide whether $f$ satisfies the property or is far' from satisfying
the property.
It has been shown that regular languages are testable,
and that there exist context free ... more >>>

TR04-014 | 26th November 2003
Chris Pollett

#### Languages to diagonalize against advice classes

Variants of Kannan's Theorem are given where the circuits of
the original theorem are replaced by arbitrary recursively presentable
classes of languages that use advice strings and satisfy certain mild
conditions. These variants imply that $\DTIME(n^{k'})^{\NE}/n^k$
does not contain $\PTIME^{\NE}$, $\DTIME(2^{n^{k'}})/n^k$ does
not contain $\EXP$, $\SPACE(n^{k'})/n^k$ does not ... more >>>

TR06-117 | 31st August 2006
Arkadev Chattopadhyay, Michal Koucky, Andreas Krebs, Mario Szegedy, Pascal Tesson, Denis Thérien

#### Languages with Bounded Multiparty Communication Complexity

We study languages with bounded communication complexity in the multiparty "input on the forehead" model with worst-case partition. In the two party case, it is known that such languages are exactly those that are recognized by programs over commutative monoids. This can be used to show that these languages can ... more >>>

TR12-052 | 27th April 2012

#### Languages with Efficient Zero-Knowledge PCPs are in SZK

A Zero-Knowledge PCP (ZK-PCP) is a randomized PCP such that the view of any (perhaps cheating) efficient verifier can be efficiently simulated up to small statistical distance. Kilian, Petrank, and Tardos (STOC '97) constructed ZK-PCPs for all languages in $NEXP$. Ishai, Mahmoody, and Sahai (TCC '12), motivated by cryptographic applications, ... more >>>

TR19-068 | 27th April 2019
Shuo Pang

#### LARGE CLIQUE IS HARD ON AVERAGE FOR RESOLUTION

Revisions: 1

We prove resolution lower bounds for $k$-Clique on the Erdos-Renyi random graph $G(n,n^{-{2\xi}\over{k-1}})$ (where $\xi>1$ is constant). First we show for $k=n^{c_0}$, $c_0\in(0,1/3)$, an $\exp({\Omega(n^{(1-\epsilon)c_0})})$ average lower bound on resolution where $\epsilon$ is arbitrary constant.

We then propose the model of $a$-irregular resolution. Extended from regular resolution, this model ... more >>>

TR12-042 | 17th April 2012
Chris Beck, Russell Impagliazzo, Shachar Lovett

#### Large Deviation Bounds for Decision Trees and Sampling Lower Bounds for AC0-circuits

There has been considerable interest lately in the complexity of distributions. Recently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is inverse-polynomially close to one. That ... more >>>

TR11-066 | 25th April 2011
Venkatesan Guruswami, Ali Kemal Sinop

#### Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Quadratic Integer Programming with PSD Objectives

Revisions: 1

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>

TR12-089 | 7th July 2012
Meena Boppana

#### Lattice Variant of the Sensitivity Conjecture

The Sensitivity Conjecture, posed in 1994, states that the fundamental measures known as the sensitivity and block sensitivity of a Boolean function $f$, $s(f)$ and $bs(f)$ respectively, are polynomially related. It is known that $bs(f)$ is polynomially related to important measures in computer science including the decision-tree depth, polynomial degree, ... more >>>

TR06-147 | 27th November 2006
Chris Peikert, Alon Rosen

#### Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors

Revisions: 1

We demonstrate an \emph{average-case} problem which is as hard as
finding $\gamma(n)$-approximate shortest vectors in certain
$n$-dimensional lattices in the \emph{worst case}, where $\gamma(n) = O(\sqrt{\log n})$. The previously best known factor for any class
of lattices was $\gamma(n) = \tilde{O}(n)$.

To obtain our ... more >>>

TR04-113 | 19th November 2004
Mårten Trolin

#### Lattices with Many Cycles Are Dense

We give a method for approximating any $n$-dimensional
lattice with a lattice $\Lambda$ whose factor group
$\mathbb{Z}^n / \Lambda$ has $n-1$ cycles of equal length
with arbitrary precision. We also show that a direct
consequence of this is that the Shortest Vector Problem and the Closest
Vector Problem cannot ... more >>>

TR21-136 | 13th September 2021
Gil Cohen, Tal Yankovitz

#### LCC and LDC: Tailor-made distance amplification and a refined separation

The Alon-Edmonds-Luby distance amplification procedure (FOCS 1995) is an algorithm that transforms a code with vanishing distance to a code with constant distance. AEL was invoked by Kopparty, Meir, Ron-Zewi, and Saraf (J. ACM 2017) for obtaining their state-of-the-art LDC, LCC and LTC. Cohen and Yankovitz (CCC 2021) devised a ... more >>>

TR19-122 | 13th September 2019
Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, Mary Wootters

#### LDPC Codes Achieve List-Decoding Capacity

Revisions: 3

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieve list-decoding capacity. These are the first graph-based codes shown to have this property. Previously, the only codes known to achieve list-decoding capacity were completely random codes, random linear codes, and codes constructed by algebraic (rather than combinatorial) techniques. ... more >>>

TR20-060 | 23rd April 2020
Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li

#### Leakage-Resilient Extractors and Secret-Sharing against Bounded Collusion Protocols

In a recent work, Kumar, Meka, and Sahai (FOCS 2019) introduced the notion of bounded collusion protocols (BCPs), in which $N$ parties wish to compute some joint function $f:(\{0,1\}^n)^N\to\{0,1\}$ using a public blackboard, but such that only $p$ parties may collude at a time. This generalizes well studied models in ... more >>>

TR22-113 | 11th August 2022
Yanyi Liu, Rafael Pass

#### Leakage-Resilient Hardness v.s. Randomness

Revisions: 1

A central open problem in complexity theory concerns the question of
whether all efficient randomized algorithms can be simulated by
efficient deterministic algorithms. The celebrated hardness
v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84),
Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness
assumptions under which $\prBPP = \prP$, but these hardness ... more >>>

TR14-129 | 10th October 2014
Divesh Aggarwal, Stefan Dziembowski, Tomasz Kazana , Maciej Obremski

#### Leakage-resilient non-malleable codes

Revisions: 1

A recent trend in cryptography is to construct cryptosystems that are secure against physical attacks. Such attacks are usually divided into two classes: the \emph{leakage} attacks in which the adversary obtains some information about the internal state of the machine, and the \emph{tampering} attacks where the adversary can modify this ... more >>>

TR18-200 | 29th November 2018
Ashutosh Kumar, Raghu Meka, Amit Sahai

#### Leakage-Resilient Secret Sharing

In this work, we consider the natural goal of designing secret sharing schemes that ensure security against a powerful adaptive adversary who may learn some leaked'' information about all the shares. We say that a secret sharing scheme is $p$-party leakage-resilient, if the secret remains statistically hidden even after an ... more >>>

TR21-095 | 8th July 2021
Marco Carmosino, Valentine Kabanets, Antonina Kolokolova, Igor Oliveira

#### LEARN-Uniform Circuit Lower Bounds and Provability in Bounded Arithmetic

We investigate randomized LEARN-uniformity, which captures the power of randomness and equivalence queries (EQ) in the construction of Boolean circuits for an explicit problem. This is an intermediate notion between P-uniformity and non-uniformity motivated by connections to learning, complexity, and logic. Building on a number of techniques, we establish the ... more >>>

TR10-074 | 20th April 2010
Parikshit Gopalan, Rocco Servedio

#### Learning and Lower Bounds for AC$^0$ with Threshold Gates

In 2002 Jackson et al. [JKS02] asked whether AC0 circuits augmented with a threshold gate at the output can be efficiently learned from uniform random examples. We answer this question affirmatively by showing that such circuits have fairly strong Fourier concentration; hence the low-degree algorithm of Linial, Mansour and Nisan ... more >>>

TR14-144 | 30th October 2014
Eric Blais, Clement Canonne, Igor Carboni Oliveira, Rocco Servedio, Li-Yang Tan

#### Learning circuits with few negations

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and ... more >>>

TR21-155 | 13th November 2021
Vishwas Bhargava, Ankit Garg, Neeraj Kayal, Chandan Saha

#### Learning generalized depth-three arithmetic circuits in the non-degenerate case

Revisions: 1

Consider a homogeneous degree $d$ polynomial $f = T_1 + \cdots + T_s$, $T_i = g_i(\ell_{i,1}, \ldots, \ell_{i, m})$ where $g_i$'s are homogeneous $m$-variate degree $d$ polynomials and $\ell_{i,j}$'s are linear polynomials in $n$ variables. We design a (randomized) learning algorithm that given black-box access to $f$, computes black-boxes for ... more >>>

TR11-115 | 8th August 2011

#### Learning Hurdles for Sleeping Experts

We study the online decision problem where the set of available actions varies over time, also called the sleeping experts problem. We consider the setting where the performance comparison is made with respect to the best ordering of actions in hindsight. In this paper, both the payoff function and the ... more >>>

TR05-088 | 3rd August 2005
Jan Arpe

#### Learning Juntas in the Presence of Noise

The combination of two major challenges in machine learning is investigated: dealing with large amounts of irrelevant information and learning from noisy data. It is shown that large classes of Boolean concepts that depend on a small number of variables---so-called juntas---can be learned efficiently from random examples corrupted by random ... more >>>

TR96-008 | 22nd January 1996
F. Bergadano, N.H. Bshouty, Stefano Varricchio

#### Learning Multivariate Polynomials from Substitution and Equivalence Queries

It has been shown in previous recent work that
multiplicity automata are predictable from multiplicity
and equivalence queries. In this paper we generalize
related notions in a matrix representation
and obtain a basis for the solution
of a number of open problems in learnability theory.
Membership queries are generalized ... more >>>

TR00-069 | 14th July 2000
Peter Auer

#### Learning Nested Differences in the Presence of Malicious Noise

We present a PAC-learning algorithm and an on-line learning algorithm
for nested differences of intersection-closed classes. Examples of
intersection-closed classes include axis-parallel rectangles,
monomials, and linear sub-spaces. Our PAC-learning algorithm uses a
pruning technique that we rigorously proof correct. As a result we
show that ... more >>>

TR00-055 | 14th July 2000
Peter Auer, Stephen Kwek, Manfred K. Warmuth

#### Learning of Depth Two Neural Networks with Constant Fan-in at the Hidden Nodes

We present algorithms for learning depth two neural networks where the
hidden nodes are threshold gates with constant fan-in. The transfer
function of the output node might be more general: we have results for
the cases when the threshold function, the logistic function or the
identity function is ... more >>>

TR09-060 | 4th June 2009
Harry Buhrman, David García Soriano, Arie Matsliah

#### Learning parities in the mistake-bound model.

We study the problem of learning parity functions that depend on at most $k$ variables ($k$-parities) attribute-efficiently in the mistake-bound model.
We design simple, deterministic, polynomial-time algorithms for learning $k$-parities with mistake bound $O(n^{1-\frac{c}{k}})$, for any constant $c > 0$. These are the first polynomial-time algorithms that learn $\omega(1)$-parities in ... more >>>

TR10-066 | 14th April 2010
Sanjeev Arora, Rong Ge

#### Learning Parities with Structured Noise

Revisions: 1

In the {\em learning parities with noise} problem ---well-studied in learning theory and cryptography--- we
have access to an oracle that, each time we press a button,
returns a random vector $a \in \GF(2)^n$ together with a bit $b \in \GF(2)$ that was computed as
$a\cdot u +\eta$, where ... more >>>

TR98-060 | 8th October 1998
Oded Goldreich, Ronitt Rubinfeld, Madhu Sudan

#### Learning polynomials with queries -- The highly noisy case.

This is a revised version of work which has appeared
in preliminary form in the 36th FOCS, 1995.

Given a function $f$ mapping $n$-variate inputs from a finite field
$F$ into $F$,
we consider the task of reconstructing a list of all $n$-variate
degree $d$ polynomials which agree with $f$
more >>>

TR07-129 | 25th October 2007
Jeffrey C. Jackson, Homin Lee, Rocco Servedio, Andrew Wan

#### Learning Random Monotone DNF

We give an algorithm that with high probability properly learns random monotone t(n)-term
DNF under the uniform distribution on the Boolean cube {0, 1}^n. For any polynomially bounded function t(n) <= poly(n) the algorithm runs in time poly(n, 1/eps) and with high probability outputs an eps accurate monotone DNF ... more >>>

TR17-046 | 8th March 2017
Sebastian Berndt, Maciej Li\'skiewicz, Matthias Lutter, Rüdiger Reischuk

#### Learning Residual Alternating Automata

Residuality plays an essential role for learning finite automata.
While residual deterministic and nondeterministic
automata have been understood quite well, fundamental
questions concerning alternating automata (AFA) remain open.
Recently, Angluin, Eisenstat, and Fisman have initiated
a systematic study of residual AFAs and proposed an algorithm called AL*
-an extension of ... more >>>

TR20-045 | 15th April 2020
Ankit Garg, Neeraj Kayal, Chandan Saha

#### Learning sums of powers of low-degree polynomials in the non-degenerate case

Revisions: 1

We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an $n$-variate degree-$d$ polynomial $f$ which can be written as
$$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$
where each $c_i\in \mathbb{F}^{\times}$, $Q_i$ is a homogeneous polynomial of degree $t$, and $t m = ... more >>> TR22-164 | 20th November 2022 Shuichi Hirahara, Mikito Nanashima #### Learning versus Pseudorandom Generators in Constant Parallel Time A polynomial-stretch pseudorandom generator (PPRG) in NC$^0$(i.e., constant parallel time) is one of the most important cryptographic primitives, especially for constructing highly efficient cryptography and indistinguishability obfuscation. The celebrated work (Applebaum, Ishai, and Kushilevitz, SIAM Journal on Computing, 2006) on randomized encodings yields the characterization of sublinear-stretch pseudorandom generators ... more >>> TR10-075 | 22nd April 2010 Ben Reichardt #### Least span program witness size equals the general adversary lower bound on quantum query complexity Span programs form a linear-algebraic model of computation, with span program "size" used in proving classical lower bounds. Quantum query complexity is a coherent generalization, for quantum algorithms, of classical decision-tree complexity. It is bounded below by a semi-definite program (SDP) known as the general adversary bound. We connect these ... more >>> TR09-062 | 28th July 2009 Daniele Venturi #### Lecture Notes on Algorithmic Number Theory The principal aim of this notes is to give a survey on the state of the art of algorithmic number theory, with particular focus on the theory of elliptic curves. Computational security is the goal of modern cryptographic constructions: the security of modern criptographic schemes stems from the assumption ... more >>> TR06-077 | 12th June 2006 Maria Lopez-Valdes #### Lempel-Ziv Dimension for Lempel-Ziv Compression This paper describes the Lempel-Ziv dimension (Hausdorff like dimension inspired in the LZ78 parsing), its fundamental properties and relation with Hausdorff dimension. It is shown that in the case of individual infinite sequences, the Lempel-Ziv dimension matches with the asymptotical Lempel-Ziv compression ratio. This fact is used to describe results ... more >>> TR05-055 | 19th May 2005 Bruno Codenotti, Amin Saberi, Kasturi Varadarajan, Yinyu Ye #### Leontief Economies Encode Nonzero Sum Two-Player Games We give a reduction from any two-player game to a special case of the Leontief exchange economy, with the property that the Nash equilibria of the game and the equilibria of the market are in one-to-one correspondence. Our reduction exposes a potential hurdle inherent in solving certain families of market ... more >>> TR11-168 | 9th December 2011 Joshua Grochow #### Lie algebra conjugacy We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set$\mathcal{L}$of matrices such that$A,B \in \mathcal{L}$implies$AB - BA \in ... more >>>

TR11-097 | 7th July 2011
Thomas Watson

#### Lift-and-Project Integrality Gaps for the Traveling Salesperson Problem

Revisions: 1

We study the lift-and-project procedures of Lovasz-Schrijver and Sherali-Adams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integrality gap of the standard relaxation is at least 2. We ... more >>>

TR17-165 | 3rd November 2017
Toniann Pitassi, Robert Robere

#### Lifting Nullstellensatz to Monotone Span Programs over Any Field

We characterize the size of monotone span programs computing certain "structured" boolean functions by the Nullstellensatz degree of a related unsatisfiable Boolean formula.

This yields the first exponential lower bounds for monotone span programs over arbitrary fields, the first exponential separations between monotone span programs over fields of different ... more >>>

TR16-048 | 11th March 2016
Olaf Beyersdorff, Leroy Chew, Renate Schmidt, Martin Suda

#### Lifting QBF Resolution Calculi to DQBF

We examine the existing Resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems Q-Resolution < IR-calc < IRM-calc, the ... more >>>

TR17-054 | 22nd March 2017
Anurag Anshu, Naresh Goud, Rahul Jain, Srijita Kundu, Priyanka Mukhopadhyay

#### Lifting randomized query complexity to randomized communication complexity

Revisions: 4

We show that for any (partial) query function $f:\{0,1\}^n\rightarrow \{0,1\}$, the randomized communication complexity of $f$ composed with $\mathrm{Index}^n_m$ (with $m= \poly(n)$) is at least the randomized query complexity of $f$ times $\log n$. Here $\mathrm{Index}_m : [m] \times \{0,1\}^m \rightarrow \{0,1\}$ is defined as $\mathrm{Index}_m(x,y)= y_x$ (the $x$th bit ... more >>>

TR18-175 | 23rd October 2018

#### Lifting Theorems for Equality

Revisions: 2

We show a deterministic simulation (or lifting) theorem for composed problems $f \circ EQ_n$ where the inner function (the gadget) is Equality on $n$ bits. When $f$ is a total function on $p$ bits, it is easy to show via a rank argument that the communication complexity of $f\circ EQ_n$ ... more >>>

TR22-172 | 2nd December 2022

#### Lifting to Parity Decision Trees Via Stifling

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple ... more >>>

TR19-186 | 31st December 2019
Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, Susanna de Rezende

#### Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity

Revisions: 4

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open ... more >>>

TR20-111 | 24th July 2020
Ian Mertz, Toniann Pitassi

#### Lifting: As Easy As 1,2,3

Revisions: 1

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Whereas previous proofs used sophisticated Fourier analytic techniques, our proof uses elementary counting together with ... more >>>

TR10-123 | 4th August 2010
Eli Ben-Sasson

#### Limitation on the rate of families of locally testable codes

Revisions: 1

This paper describes recent results which revolve around the question of the rate attainable by families of error correcting codes that are locally testable. Emphasis is placed on motivating the problem of proving upper bounds on the rate of these codes and a number of interesting open questions for future ... more >>>

TR08-007 | 6th February 2008

#### Limitations of Hardness vs. Randomness under Uniform Reductions

We consider (uniform) reductions from computing a function f to the task of distinguishing the output of some pseudorandom generator G from uniform. Impagliazzo and Wigderson (FOCS 98, JCSS 01) and Trevisan and Vadhan (CCC 02, CC 07) exhibited such reductions for every function f in PSPACE. Moreover, their reductions ... more >>>

TR12-041 | 17th April 2012
Stasys Jukna

#### Limitations of Incremental Dynamic Programs

Revisions: 1

We consider so-called incremental'' dynamic programming (DP) algorithms, and are interested in the number of subproblems produced by them. The standard DP algorithm for the n-dimensional Knapsack problem is incremental, and produces nK subproblems, where K is the capacity of the knapsack. We show that any incremental algorithm for this ... more >>>

TR16-017 | 24th December 2015
Georgios Stamoulis

#### Limitations of Linear Programming Techniques for Bounded Color Matchings

Given a weighted graph $G = (V,E,w)$, with weight function $w: E \rightarrow \mathbb{Q^+}$, a \textit{matching} $M$ is a set of pairwise non-adjacent edges. In the optimization setting, one seeks to find a matching of \textit{maximum} weight. In the \textit{multi-criteria} (or \textit{multi-budgeted}) setting, we are also given $\ell$ length functions ... more >>>

TR12-075 | 12th June 2012

#### Limitations of Local Filters of Lipschitz and Monotone Functions

We study local filters for two properties of functions $f:\B^d\to \mathbb{R}$: the Lipschitz property and monotonicity. A local filter with additive error $a$ is a randomized algorithm that is given black-box access to a function $f$ and a query point $x$ in the domain of $f$. Its output is a ... more >>>

TR11-125 | 16th September 2011
Andrew Drucker

#### Limitations of Lower-Bound Methods for the Wire Complexity of Boolean Operators

We study the circuit complexity of Boolean operators, i.e., collections of Boolean functions defined over a common input. Our focus is the well-studied model in which arbitrary Boolean functions are allowed as gates, and in which a circuit's complexity is measured by its depth and number of wires. We show ... more >>>

TR04-026 | 17th February 2004
Scott Aaronson

#### Limitations of Quantum Advice and One-Way Communication

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
First, we show that BQP/qpoly is contained in ... more >>>

TR15-201 | 10th December 2015
C Ramya, Raghavendra Rao B V

#### Limitations of sum of products of Read-Once Polynomials

Revisions: 1

We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas.
We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a product of variable ... more >>>

TR21-108 | 22nd July 2021

#### Limitations of the Impagliazzo--Nisan--Wigderson Pseudorandom Generator against Permutation Branching Programs

The classic Impagliazzo--Nisan--Wigderson (INW) psesudorandom generator (PRG) (STOC 94) for space-bounded computation uses a seed of length $O(\log n \cdot \log(nwd/\varepsilon))$ to fool ordered branching programs of length $n$, width $w$, and alphabet size $d$ to within error $\varepsilon$. A series of works have shown that the analysis of the ... more >>>

TR14-067 | 4th May 2014
Venkatesan Guruswami, Madhu Sudan, Ameya Velingker, Carol Wang

#### Limitations on Testable Affine-Invariant Codes in the High-Rate Regime

Locally testable codes (LTCs) of constant distance that allow the tester to make a linear number of queries have become the focus of attention recently, due to their elegant connections to hardness of approximation. In particular, the binary Reed-Muller code of block length $N$ and distance $d$ is known to ... more >>>

TR13-055 | 5th April 2013

#### Limits of local algorithms over sparse random graphs

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Recent research ... more >>>

TR15-198 | 30th November 2015
Shuichi Hirahara, Osamu Watanabe

#### Limits of Minimum Circuit Size Problem as Oracle

Revisions: 1

The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP $\neq$ EXP, which is a ... more >>>

TR12-156 | 12th November 2012
Andrej Bogdanov, Chin Ho Lee

#### Limits of provable security for homomorphic encryption

Revisions: 1

We show that public-key bit encryption schemes which support weak homomorphic evaluation of parity or majority cannot be proved message indistinguishable beyond AM intersect coAM via general (adaptive) reductions, and beyond statistical zero-knowledge via reductions of constant query complexity.

Previous works on the limitation of reductions for proving security of ... more >>>

TR12-065 | 16th May 2012
Mohammad Mahmoody, Hemanta Maji, Manoj Prabhakaran

#### Limits of Random Oracles in Secure Computation

Revisions: 2

The seminal result of Impagliazzo and Rudich (STOC 1989) gave a black-box separation between one-way functions and public-key encryption: informally, a public-key encryption scheme cannot be constructed using one-way functions as the sole source of computational hardness. In addition, this implied a black-box separation between one-way functions and protocols for ... more >>>

TR23-008 | 2nd February 2023
Ond?ej Ježil

#### Limits of structures and Total NP Search Problems

For a class of finite graphs, we define a limit object relative to some computationally restricted class of functions. The properties of the limit object then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses Krají?ek's forcing with random variables [7]. We prove ... more >>>

TR11-031 | 8th March 2011
Sam Buss, Ryan Williams

#### Limits on Alternation-Trading Proofs for Time-Space Lower Bounds

This paper characterizes alternation trading based proofs that satisfiability is not in the time and space bounded class $\DTISP(n^c, n^\epsilon)$, for various values $c<2$ and $\epsilon<1$. We characterize exactly what can be proved in the $\epsilon=0$ case with currently known methods, and prove the conjecture of Williams that $c=2\cos(\pi/7)$ is ... more >>>

TR17-060 | 9th April 2017
Boaz Barak, Zvika Brakerski, Ilan Komargodski, Pravesh Kothari

#### Limits on Low-Degree Pseudorandom Generators (Or: Sum-of-Squares Meets Program Obfuscation)

Revisions: 1

We prove that for every function $G\colon\{0,1\}^n \rightarrow \mathbb{R}^m$, if every output of $G$ is a polynomial (over $\mathbb{R}$) of degree at most $d$ of at most $s$ monomials and $m > \widetilde{O}(sn^{\lceil d/2 \rceil})$, then there is a polynomial time algorithm that can distinguish a vector of the form ... more >>>

TR10-139 | 17th September 2010
Eric Allender, Luke Friedman, William Gasarch

#### Limits on the Computational Power of Random Strings

Revisions: 1

Let C(x) and K(x) denote plain and prefix Kolmogorov complexity, respectively, and let R_C and R_K denote the sets of strings that are random'' according to these measures; both R_K and R_C are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both R_K ... more >>>

TR06-148 | 4th December 2006
Chris Peikert

#### Limits on the Hardness of Lattice Problems in $\ell_p$ Norms

Revisions: 1

We show that for any $p \geq 2$, lattice problems in the $\ell_p$
norm are subject to all the same limits on hardness as are known
for the $\ell_2$ norm. In particular, for lattices of dimension
$n$:

* Approximating the shortest and closest vector in ... more >>>

TR10-108 | 9th July 2010

#### Limits on the rate of locally testable affine-invariant codes

A linear code is said to be affine-invariant if the coordinates of the code can be viewed as a vector space and the code is invariant under an affine transformation of the coordinates. A code is said to be locally testable if proximity of a received word
to the code ... more >>>

TR09-068 | 1st September 2009
Dave Buchfuhrer, Chris Umans

#### Limits on the Social Welfare of Maximal-In-Range Auction Mechanisms

Many commonly-used auction mechanisms are maximal-in-range''. We show that any maximal-in-range mechanism for $n$ bidders and $m$ items cannot both approximate the social welfare with a ratio better than $\min(n, m^\eta)$ for any constant $\eta < 1/2$ and run in polynomial time, unless $NP \subseteq P/poly$. This significantly improves upon ... more >>>

TR09-013 | 4th February 2009
Atri Rudra

#### Limits to List Decoding Random Codes

It has been known since [Zyablov and Pinsker 1982] that a random $q$-ary code of rate $1-H_q(\rho)-\eps$ (where $0<\rho<1-1/q$, $\eps>0$ and $H_q(\cdot)$ is the $q$-ary entropy function) with high probability is a $(\rho,1/\eps)$-list decodable code. (That is, every Hamming ball of radius at most $\rho n$ has at most $1/\eps$ ... more >>>

TR96-005 | 9th January 1996
Hans-Joerg Burtschick, Heribert Vollmer

#### Lindstroem Quantifiers and Leaf Language Definability

Revisions: 1

We show that examinations of the expressive power of logical formulae
enriched by Lindstroem quantifiers over ordered finite structures
have a well-studied complexity-theoretic counterpart: the leaf
language approach to define complexity classes. Model classes of
formulae with Lindstroem quantifiers are nothing else than leaf
language definable sets. Along the ... more >>>

TR05-042 | 15th April 2005

#### Linear Advice for Randomized Logarithmic Space

Revisions: 1

We show that RL is contained in L/O(n), i.e., any language computable
in randomized logarithmic space can be computed in deterministic
logarithmic space with a linear amount of non-uniform advice. To
prove our result we show how to take an ultra-low space walk on
the Gabber-Galil expander graph.

more >>>

TR01-074 | 12th October 2001
Joshua Buresh-Oppenheim, David Mitchell

#### Linear and Negative Resolution are Weaker than Resolution

We prove exponential separations between the sizes of
particular refutations in negative, respectively linear, resolution and
general resolution. Only a superpolynomial separation between negative
and general resolution was previously known. Our examples show that there
is no strong relationship between the size and width of refutations in
negative and ... more >>>

TR07-052 | 7th May 2007
Li Chen, Bin Fu

#### Linear and Sublinear Time Algorithms for the Basis of Abelian Groups

Revisions: 2

It is well known that every finite Abelian group $G$ can be
represented as a product of cyclic groups: $G=G_1\times G_2\times\cdots G_t$, where each $G_i$ is a cyclic group of size
$p^j$ for some prime $p$ and integer $j\ge 1$. If $a_i$ is the
generator of the cyclic group of ... more >>>

TR14-141 | 24th October 2014
Shachar Lovett

#### Linear codes cannot approximate the network capacity within any constant factor

Network coding studies the capacity of networks to carry information, when internal nodes are allowed to actively encode information. It is known that for multi-cast networks, the network coding capacity can be achieved by linear codes. It is also known not to be true for general networks. The best separation ... more >>>

TR99-025 | 2nd July 1999

#### Linear Consistency Testing

We extend the notion of linearity testing to the task of checking
linear-consistency of multiple functions. Informally, functions
are linear'' if their graphs form straight lines on the plane.
Two such functions are consistent'' if the lines have the same
slope. We propose a variant of a test of ... more >>>

TR05-100 | 30th August 2005
David Zuckerman

#### Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number

A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. We construct new extractors which require only log n + O(1) additional random bits for sources with constant entropy rate. We further construct dispersers, which are similar to one-sided extractors, ... more >>>

TR22-047 | 4th April 2022
Manik Dhar, Zeev Dvir

#### Linear Hashing with $\ell_\infty$ guarantees and two-sided Kakeya bounds

Revisions: 1

We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $|S|$. Let ... more >>>

TR15-017 | 20th January 2015
Bruno Bauwens, Marius Zimand

#### Linear list-approximation for short programs (or the power of a few random bits)

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that contains a $O(\log n)$-short program for $x$. ... more >>>

TR16-182 | 14th November 2016
Rohit Gurjar, Thomas Thierauf

#### Linear Matroid Intersection is in quasi-NC

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. We show that the linear matroid intersection problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(\log^2 n)$ depth. This generalizes ... more >>>

TR07-033 | 14th February 2007
Michael Navon, Alex Samorodnitsky

#### Linear programming bounds for codes via a covering argument

We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, ... more >>>

TR17-086 | 9th May 2017
C Ramya, Raghavendra Rao B V

#### Linear Projections of the Vandermonde Polynomial

Revisions: 1

An n-variate Vandermonde polynomial is the determinant of the n × n matrix where the ith column is the vector (1, x_i , x_i^2 , . . . , x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation ... more >>>

TR16-174 | 7th November 2016
Elchanan Mossel, Sampath Sampath Kannan, Grigory Yaroslavtsev

#### Linear Sketching over $\mathbb F_2$

We initiate a systematic study of linear sketching over $\mathbb F_2$. For a given Boolean function $f \colon \{0,1\}^n \to \{0,1\}$ a randomized $\mathbb F_2$-sketch is a distribution $\mathcal M$ over $d \times n$ matrices with elements over $\mathbb F_2$ such that $\mathcal Mx$ suffices for computing $f(x)$ with high ... more >>>

TR21-086 | 22nd June 2021
Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Ameya Velingker, Santhoshini Velusamy

#### Linear Space Streaming Lower Bounds for Approximating CSPs

Revisions: 1

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify ... more >>>

TR11-048 | 10th April 2011

#### Linear systems over abelian groups

We consider a system of linear constraints over any finite Abelian group $G$ of the following form: $\ell_i(x_1,\ldots,x_n) \equiv \ell_{i,1}x_1+\cdots+\ell_{i,n}x_n \in A_i$ for $i=1,\ldots,t$ and each $A_i \subset G$, $\ell_{i,j}$ is an element of $G$ and $x_i$'s are Boolean variables. Our main result shows that the subset of the Boolean ... more >>>

TR09-084 | 24th September 2009

#### Linear systems over composite moduli

We study solution sets to systems of generalized linear equations of the following form:
$\ell_i (x_1, x_2, \cdots , x_n)\, \in \,A_i \,\, (\text{mod } m)$,
where $\ell_1, \ldots ,\ell_t$ are linear forms in $n$ Boolean variables, each $A_i$ is an arbitrary subset of $\mathbb{Z}_m$, and $m$ is a composite ... more >>>

TR23-012 | 16th February 2023
Yogesh Dahiya, Vignesh K, Meena Mahajan, Karteek Sreenivasaiah

#### Linear threshold functions in decision lists, decision trees, and depth-2 circuits

We show that polynomial-size constant-rank linear decision trees (LDTs) can be converted to polynomial-size depth-2 threshold circuits LTF$\circ$LTF. An intermediate construct is polynomial-size decision lists that query a conjunction of a constant number of linear threshold functions (LTFs); we show that these are equivalent to polynomial-size exact linear decision lists ... more >>>

TR21-050 | 2nd April 2021
Marshall Ball, Alper Cakan, Tal Malkin

#### Linear Threshold Secret-Sharing with Binary Reconstruction

Motivated in part by applications in lattice-based cryptography, we initiate the study of the size of linear threshold ($t$-out-of-$n$') secret-sharing where the linear reconstruction function is restricted to coefficients in $\{0,1\}$. We prove upper and lower bounds on the share size of such schemes. One ramification of our results is ... more >>>

TR08-101 | 20th November 2008
Marek Karpinski, Warren Schudy

#### Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems

We design a linear time approximation scheme for the Gale-Berlekamp Switching Game and generalize it to a wider class of dense fragile minimization problems including the Nearest Codeword Problem (NCP) and Unique Games Problem. Further applications include, among other things, finding a constrained form of matrix rigidity and maximum likelihood ... more >>>

TR11-058 | 15th April 2011
Michael Viderman

#### Linear time decoding of regular expander codes

Revisions: 1

Sipser and Spielman (IEEE IT, 1996) showed that any $(c,d)$-regular expander code with expansion parameter $> \frac{3}{4}$ is decodable in \emph{linear time} from a constant fraction of errors. Feldman et al. (IEEE IT, 2007)
proved that expansion parameter $> \frac{2}{3} + \frac{1}{3c}$ is sufficient to correct a constant fraction of ... more >>>

TR04-016 | 3rd March 2004
Michael Alekhnovich, Eli Ben-Sasson

#### Linear Upper Bounds for Random Walk on Small Density Random 3CNFs

We analyze the efficiency of the random walk algorithm on random 3CNF instances, and prove em linear upper bounds on the running time
of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the ... more >>>

TR12-073 | 11th June 2012
Venkatesan Guruswami, Carol Wang

#### Linear-algebraic list decoding for variants of Reed-Solomon codes

Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and list error-correction capability. Specifically, for any $\epsilon > 0$, Guruswami and Rudra presented an $n^{O(1/\epsilon)}$ time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction $1-R-\epsilon$ of ... more >>>

TR20-013 | 17th February 2020
Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor

Revisions: 1

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $\cC_0$ of length $d$, and a $d$-regular bipartite expander graph $G$ with $n$ vertices on each side, we give an algorithm to list-decode the expander code $\cC ... more >>> TR14-015 | 24th January 2014 Jack H. Lutz, Neil Lutz #### Lines Missing Every Random Point Revisions: 1 This paper proves that there is, in every direction in Euclidean space, a line that misses every computably random point. Our proof of this fact shows that a famous set constructed by Besicovitch in 1964 has computable measure 0. more >>> TR14-007 | 17th January 2014 Mark Braverman, Klim Efremenko #### List and Unique Coding for Interactive Communication in the Presence of Adversarial Noise In this paper we extend the notion of list decoding to the setting of interactive communication and study its limits. In particular, we show that any protocol can be encoded, with a constant rate, into a list-decodable protocol which is resilient to a noise rate of up to$1/2-\varepsilon$, ... more >>> TR11-165 | 8th December 2011 Elena Grigorescu, Chris Peikert #### List Decoding Barnes-Wall Lattices Revisions: 2 The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in$R^{N}$, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the ... more >>> TR14-087 | 12th July 2014 Abhishek Bhowmick, Shachar Lovett #### List decoding Reed-Muller codes over small fields Revisions: 1 The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes. Fix a finite field$\mathbb{F}$. ... more >>> TR12-146 | 7th November 2012 Venkatesan Guruswami, Chaoping Xing #### List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code ... more >>> TR08-105 | 26th November 2008 Parikshit Gopalan, Venkatesan Guruswami, Prasad Raghavendra, Prasad Raghavendra #### List Decoding Tensor Products and Interleaved Codes We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. 1)We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one ... more >>> TR03-042 | 15th May 2003 Luca Trevisan #### List Decoding Using the XOR Lemma We show that Yao's XOR Lemma, and its essentially equivalent rephrasing as a Direct Product Lemma, can be re-interpreted as a way of obtaining error-correcting codes with good list-decoding algorithms from error-correcting codes having weak unique-decoding algorithms. To get codes with good rate and efficient list decoding algorithms one needs ... more >>> TR18-136 | 1st August 2018 Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, Amnon Ta-Shma #### List Decoding with Double Samplers Revisions: 1 We develop the notion of double samplers, first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders. We show how double samplers give a generic way of amplifying distance in a way that enables ... more >>> TR02-024 | 24th April 2002 Piotr Indyk #### List-decoding in Linear Time Spielman showed that one can construct error-correcting codes capable of correcting a constant fraction$\delta << 1/2$of errors, and that are encodable/decodable in linear time. Guruswami and Sudan showed that it is possible to correct more than$50\%$of errors (and thus exceed the half of the ... more >>> TR12-044 | 22nd April 2012 Swastik Kopparty #### List-Decoding Multiplicity Codes We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching$1$while simultaneously allowing for sublinear-time error-correction. In this paper, we show that multiplicity codes also admit powerful list-decoding and local list-decoding algorithms correcting a large fraction ... more >>> TR22-069 | 28th April 2022 Silas Richelson, Sourya Roy #### List-Decoding Random Walk XOR Codes Near the Johnson Bound Revisions: 1 In a breakthrough result, Ta-Shma described an explicit construction of an almost optimal binary code (STOC 2017). Ta-Shma's code has distance$\frac{1-\varepsilon}{2}$and rate$\Omega\bigl(\varepsilon^{2+o(1)}\bigr)$and thus it almost achieves the Gilbert-Varshamov bound, except for the$o(1)$term in the exponent. The prior best list-decoding algorithm for (a variant of) ... more >>> TR11-020 | 20th December 2010 Yijia Chen, Joerg Flum #### Listings and logics There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In [Chen and Flum, 2010] we have shown that${\rm LFP}_{\rm inv}$, the order-invariant least fixed-point logic LFP,'' captures P (on all finite structures) if and only if there is ... more >>> TR15-038 | 11th March 2015 Gil Cohen #### Local Correlation Breakers and Applications to Three-Source Extractors and Mergers Revisions: 1 We introduce and construct a pseudorandom object which we call a local correlation breaker (LCB). Informally speaking, an LCB is a function that gets as input a sequence of$r$(arbitrarily correlated) random variables and an independent weak-source. The output of the LCB is a sequence of$r$random variables ... more >>> TR18-141 | 6th August 2018 Sandip Sinha, Omri Weinstein #### Local Decodability of the Burrows-Wheeler Transform Revisions: 1 The Burrows-Wheeler Transform (BWT) is among the most influential discoveries in text compression and DNA storage. It is a \emph{reversible} preprocessing step that rearranges an$n$-letter string into runs of identical characters (by exploiting context regularities), resulting in highly compressible strings, and is the basis for the ubiquitous \texttt{bzip} program. ... more >>> TR17-138 | 17th September 2017 Srikanth Srinivasan, Madhu Sudan #### Local decoding and testing of polynomials over grids Revisions: 1 The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that$n$-variate polynomials of total degree at most$d$over grids, i.e. sets of the form$A_1 \times A_2 \times \cdots \times A_n$, form error-correcting codes (of distance at least$2^{-d}$provided$\min_i\{|A_i|\}\geq 2$). In this work we explore their local decodability and local testability. ... more >>> TR16-129 | 16th August 2016 Emanuele Viola, Avi Wigderson #### Local Expanders Revisions: 1 Abstract A map$f:{0,1}^{n}\to {0,1}^{n}$has locality t if every output bit of f depends only on t input bits. Arora, Steurer, and Wigderson (2009) ask if there exist bounded-degree expander graphs on$2^{n}$nodes such that the neighbors of a node$x\in {0,1}^{n}$can be computed by maps of ... more >>> TR07-108 | 27th October 2007 Moses Charikar, Konstantin Makarychev, Yury Makarychev #### Local Global Tradeoffs in Metric Embeddings Suppose that every k points in a n point metric space X are D-distortion embeddable into l_1. We give upper and lower bounds on the distortion required to embed the entire space X into l_1. This is a natural mathematical question and is also motivated by the study of relaxations ... more >>> TR10-047 | 23rd March 2010 Avraham Ben-Aroya, Klim Efremenko, Amnon Ta-Shma #### Local list decoding with a constant number of queries Revisions: 1 Recently Efremenko showed locally-decodable codes of sub-exponential length. That result showed that these codes can handle up to$\frac{1}{3} $fraction of errors. In this paper we show that the same codes can be locally unique-decoded from error rate$\half-\alpha$for any$\alpha>0$and locally list-decoded from error rate$1-\alpha$... more >>> TR17-104 | 13th June 2017 Brett Hemenway, Noga Ron-Zewi, Mary Wootters #### Local List Recovery of High-rate Tensor Codes & Applications Revisions: 1 In this work, we give the first construction of {\em high-rate} locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first {\em capacity-achieving} locally list-decodable ... more >>> TR09-115 | 13th November 2009 Swastik Kopparty, Shubhangi Saraf #### Local list-decoding and testing of random linear codes from high-error In this paper, we give surprisingly efficient algorithms for list-decoding and testing {\em random} linear codes. Our main result is that random sparse linear codes are locally testable and locally list-decodable in the {\em high-error} regime with only a {\em constant} number of queries. More precisely, we show that ... more >>> TR16-163 | 25th October 2016 Matthew Hastings #### Local Maxima and Improved Exact Algorithm for MAX-2-SAT Given a MAX-2-SAT instance, we define a local maximum to be an assignment such that changing any single variable reduces the number of satisfied clauses. We consider the question of the number of local maxima hat an instance of MAX-2-SAT can have. We give upper bounds in both the sparse ... more >>> TR19-127 | 19th September 2019 Noga Ron-Zewi, Ron Rothblum #### Local Proofs Approaching the Witness Length Revisions: 1 Interactive oracle proofs (IOPs) are a hybrid between interactive proofs and PCPs. In an IOP the prover is allowed to interact with a verifier (like in an interactive proof) by sending relatively long messages to the verifier, who in turn is only allowed to query a few of the bits ... more >>> TR15-128 | 10th August 2015 Roee David, Elazar Goldenberg, Robert Krauthgamer #### Local Reconstruction of Low-Rank Matrices and Subspaces Revisions: 2 We study the problem of \emph{reconstructing a low-rank matrix}, where the input is an$n\times m$matrix$M$over a field$\mathbb{F}$and the goal is to reconstruct a (near-optimal) matrix$M'$that is low-rank and close to$M$under some distance function$\Delta$. Furthermore, the reconstruction must be local, ... more >>> TR13-099 | 6th July 2013 Hamidreza Jahanjou, Eric Miles, Emanuele Viola #### Local reductions Revisions: 3 We reduce non-deterministic time$T \ge 2^n$to a 3SAT instance$\phi$of size$|\phi| = T \cdot \log^{O(1)} T$such that there is an explicit circuit$C$that on input an index$i$of$\log |\phi|$bits outputs the$i$th clause, and each output bit of$C$depends on ... more >>> TR17-126 | 7th August 2017 Swastik Kopparty, Shubhangi Saraf #### Local Testing and Decoding of High-Rate Error-Correcting Codes We survey the state of the art in constructions of locally testable codes, locally decodable codes and locally correctable codes of high rate. more >>> TR16-125 | 31st July 2016 Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu #### Local Testing for Membership in Lattices Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in ... more >>> TR11-158 | 25th November 2011 Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, Luc Segoufin #### Locality from Circuit Lower Bounds We study the locality of an extension of first-order logic that captures graph queries computable in AC$^0$, i.e., by families of polynomial-size constant-depth circuits. The extension considers first-order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the ... more >>> TR13-098 | 28th June 2013 Benny Applebaum, Yoni Moses #### Locally Computable UOWHF with Linear Shrinkage Revisions: 2 We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs)$H:\{0,1\}^n \rightarrow \{0,1\}^m$. A construction with constant \emph{output locality}, where every bit of the output depends only on a constant number of bits of the input, was established by [Applebaum, Ishai, and Kushilevitz, SICOMP 2006]. However, this ... more >>> TR03-046 | 11th June 2003 Philippe Moser #### Locally Computed Baire's Categories on Small Complexity Classes We strengthen an earlier notion of resource-bounded Baire's categories, and define resource bounded Baire's categories on small complexity classes such as P, QP, SUBEXP and on probabilistic complexity classes such as BPP. We give an alternative characterization of meager sets via resource-bounded Banach Mazur games. We show that the class ... more >>> TR04-051 | 10th June 2004 Zdenek Dvorák, Daniel Král, Ondrej Pangrác #### Locally consistent constraint satisfaction problems An instance of a constraint satisfaction problem is$l$-consistent if any$l$constraints of it can be simultaneously satisfied. For a set$\Pi$of constraint types,$\rho_l(\Pi)$denotes the largest ratio of constraints which can be satisfied in any$l$-consistent instance composed by constraints from the set$\Pi$. In the ... more >>> TR14-107 | 10th August 2014 Or Meir #### Locally Correctable and Testable Codes Approaching the Singleton Bound Revisions: 2 Locally-correctable codes (LCCs) and locally-testable codes (LTCs) are codes that admit local algorithms for decoding and testing respectively. The local algorithms are randomized algorithms that make only a small number of queries to their input. LCCs and LTCs are both interesting in their own right, and have important applications in ... more >>> TR07-040 | 12th April 2007 Kiran Kedlaya, Sergey Yekhanin #### Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major ... more >>> TR05-044 | 6th April 2005 Zeev Dvir, Amir Shpilka #### Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits In this work we study two seemingly unrelated notions. Locally Decodable Codes(LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing ... more >>> TR13-115 | 27th August 2013 Daniele Micciancio #### Locally Dense Codes The Minimum Distance Problem (MDP), i.e., the computational task of evaluating (exactly or approximately) the minimum distance of a linear code, is a well known NP-hard problem in coding theory. A key element in essentially all known proofs that MDP is NP-hard is the construction of a combinatorial object that ... more >>> TR03-050 | 16th June 2003 Daniel Král #### Locally satisfiable formulas A CNF formula is k-satisfiable if each k clauses of it can be satisfied simultaneously. Let \pi_k be the largest real number such that for each k-satisfiable formula with variables x_i, there are probabilities p_i with the following property: If each variable x_i is chosen randomly and independently to be ... more >>> TR16-122 | 11th August 2016 Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf #### Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate$R$and minimum distance$\delta$of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every$\delta \in (0, 1/2)$, there are ... more >>> TR09-128 | 29th November 2009 Gillat Kol, Ran Raz #### Locally Testable Codes Analogues to the Unique Games Conjecture Do Not Exist The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>> TR13-114 | 24th August 2013 Parikshit Gopalan, Salil Vadhan, Yuan Zhou #### Locally Testable Codes and Cayley Graphs Revisions: 1 We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over$\F_2^h$: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over$\F_2^h$whose set of generators is significantly larger than$h$and has no short linear dependencies, but yields a ... more >>> TR02-050 | 5th August 2002 Oded Goldreich, Madhu Sudan #### Locally Testable Codes and PCPs of Almost-Linear Length Locally testable codes are error-correcting codes that admit very efficient codeword tests. Specifically, using a constant number of (random) queries, non-codewords are rejected with probability proportional to their distance from the code. Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is ... more >>> TR09-126 | 26th November 2009 Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, Michael Viderman #### Locally Testable Codes Require Redundant Testers Revisions: 3 Locally testable codes (LTCs) are error-correcting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes whose duals have (superlinearly) {\em many} small weight ... more >>> TR20-072 | 5th May 2020 Yotam Dikstein, Irit Dinur, Prahladh Harsha, Noga Ron-Zewi #### Locally testable codes via high-dimensional expanders Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are far from all codewords, by probing a given word only at a very small (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. ... more >>> TR21-151 | 8th November 2021 Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, Shahar Mozes #### Locally Testable Codes with constant rate, distance, and locality Revisions: 1 A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads$q$bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter$q$is called the locality of the tester. LTCs were initially studied as ... more >>> TR19-117 | 4th September 2019 Silas Richelson, Sourya Roy #### Locally Testable Non-Malleable Codes Revisions: 1 In this work we adapt the notion of non-malleability for codes or Dziembowski, Pietrzak and Wichs (ICS 2010) to locally testable codes. Roughly speaking, a locally testable code is non-malleable if any tampered codeword which passes the local test with good probability is close to a valid codeword which either ... more >>> TR10-130 | 18th August 2010 Tali Kaufman, Michael Viderman #### Locally Testable vs. Locally Decodable Codes Revisions: 1 We study the relation between locally testable and locally decodable codes. Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. Locally decodable codes (LDCs) allow to recover each message entry with ... more >>> TR20-155 | 18th October 2020 Alexander Knop, Shachar Lovett, Sam McGuire, Weiqiang Yuan #### Log-rank and lifting for AND-functions Revisions: 1 Let$f: \{0,1\}^n \to \{0, 1\}$be a boolean function, and let$f_\land (x, y) = f(x \land y)$denote the AND-function of$f$, where$x \land y$denotes bit-wise AND. We study the deterministic communication complexity of$f_\land$and show that, up to a$\log n$factor, it is ... more >>> TR19-149 | 4th November 2019 Dean Doron, Pooya Hatami, William Hoza #### Log-Seed Pseudorandom Generators via Iterated Restrictions There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The iterated restrictions'' approach, pioneered by Ajtai and Wigderson [AW89], has provided PRGs with seed length$\mathrm{polylog} n$or even$\tilde{O}(\log n)$for several restricted models of computation. Can this approach ever achieve the optimal seed ... more >>> TR01-013 | 2nd February 2001 Michal Koucky #### Log-space Constructible Universal Traversal Sequences for Cycles of Length$O(n^{4.03})$The paper presents a simple construction of polynomial length universal traversal sequences for cycles. These universal traversal sequences are log-space (even$NC^1$) constructible and are of length$O(n^{4.03})$. Our result improves the previously known upper-bound$O(n^{4.76})$for log-space constructible universal traversal sequences for cycles. more >>> TR07-091 | 10th September 2007 Martin Grohe #### Logic, Graphs, and Algorithms Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle's Theorem, which states that all graph properties definable in monadic second-order logic ... more >>> TR16-154 | 21st September 2016 Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, Nate Soares, Jessica Taylor #### Logical Induction We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running ... more >>> TR01-088 | 29th October 2001 Alexander Shen, Nikolay Vereshchagin #### Logical operations and Kolmogorov complexity We define Kolmogorov complexity of a set of strings as the minimal Kolmogorov complexity of its element. Consider three logical operations on sets going back to Kolmogorov and Kleene: (A OR B) is the direct sum of A,B, (A AND B) is the cartesian product of A,B, more >>> TR01-089 | 29th October 2001 Andrej Muchnik, Nikolay Vereshchagin #### Logical operations and Kolmogorov complexity. II We invistigate what is the minimal length of a program mapping A to B and at the same time mapping C to D (where A,B,C,D are binary strings). We prove that it cannot be expressed in terms of Kolmogorv complexity of A,B,C,D, their pairs (A,B), (A,C), ..., their ... more >>> TR03-077 | 4th September 2003 Till Tantau #### Logspace Optimisation Problems and their Approximation Properties This paper introduces logspace optimisation problems as analogues of the well-studied polynomial-time optimisation problems. Similarly to them, logspace optimisation problems can have vastly different approximation properties, even though the underlying existence and budget problems have the same computational complexity. Numerous natural problems are presented that exhibit such a varying ... more >>> TR09-050 | 28th May 2009 Jan Kyncl, Tomas Vyskocil #### Logspace reduction of directed reachability for bounded genus graphs to the planar case Directed reachability (or briefly reachability) is the following decision problem: given a directed graph G and two of its vertices s,t, determine whether there is a directed path from s to t in G. Directed reachability is a standard complete problem for the complexity class NL. Planar reachability is an ... more >>> TR10-062 | 7th April 2010 Michael Elberfeld, Andreas Jakoby, Till Tantau #### Logspace Versions of the Theorems of Bodlaender and Courcelle Bodlaender's Theorem states that for every$k$there is a linear-time algorithm that decides whether an input graph has tree width~$k$and, if so, computes a width-$k$tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula$\phi$and for every$k$there is ... more >>> TR96-060 | 19th November 1996 Bernd Borchert, Frank Stephan #### Looking for an Analogue of Rice's Theorem in Complexity Theory Rice's Theorem says that every nontrivial semantic property of programs is undecidable. It this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. more >>> TR18-017 | 26th January 2018 Venkatesan Guruswami, Nicolas Resch, Chaoping Xing #### Lossless dimension expanders via linearized polynomials and subspace designs For a vector space$\mathbb{F}^n$over a field$\mathbb{F}$, an$(\eta,\beta)$-dimension expander of degree$d$is a collection of$d$linear maps$\Gamma_j : \mathbb{F}^n \to \mathbb{F}^n$such that for every subspace$U$of$\mathbb{F}^n$of dimension at most$\eta n$, the image of$U$under all the maps,$\sum_{j=1}^d ... more >>>

TR07-080 | 7th August 2007
Chris Peikert, Brent Waters

#### Lossy Trapdoor Functions and Their Applications

We propose a new general primitive called lossy trapdoor
functions (lossy TDFs), and realize it under a variety of different
number theoretic assumptions, including hardness of the decisional
Diffie-Hellman (DDH) problem and the worst-case hardness of
standard lattice problems.

Using lossy TDFs, we develop a new approach for constructing ... more >>>

TR09-127 | 25th November 2009
Brett Hemenway, Rafail Ostrovsky

#### Lossy Trapdoor Functions from Smooth Homomorphic Hash Proof Systems

Revisions: 2

In STOC '08, Peikert and Waters introduced a powerful new primitive called Lossy Trapdoor Functions (LTDFs). Since their introduction, lossy trapdoor functions have found many uses in cryptography. In the work of Peikert and Waters, lossy trapdoor functions were used to give an efficient construction of a chosen-ciphertext secure ... more >>>

TR17-180 | 26th November 2017
Irit Dinur, Yuval Filmus, Prahladh Harsha

#### Low degree almost Boolean functions are sparse juntas

Nisan and Szegedy showed that low degree Boolean functions are juntas. Kindler and Safra showed that low degree functions which are *almost* Boolean are close to juntas. Their result holds with respect to $\mu_p$ for every *constant* $p$. When $p$ is allowed to be very small, new phenomena emerge. ... more >>>

TR22-051 | 18th April 2022
Vipul Arora, Arnab Bhattacharyya, Noah Fleming, Esty Kelman, Yuichi Yoshida

#### Low Degree Testing over the Reals

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast ... more >>>

TR15-111 | 8th July 2015
Diptarka Chakraborty, Elazar Goldenberg, Michal Koucky

#### Low Distortion Embedding from Edit to Hamming Distance using Coupling

Revisions: 1

The Hamming and the edit metrics are two common notions of measuring distances between pairs of strings $x,y$ lying in the Boolean hypercube. The edit distance between $x$ and $y$ is defined as the minimum number of character insertion, deletion, and bit flips needed for converting $x$ into $y$. ... more >>>

TR10-004 | 6th January 2010
Eli Ben-Sasson, Michael Viderman

#### Low Rate Is Insufficient for Local Testability

Revisions: 3

Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations.
Kaufman and Sudan \cite{KS07} proved that sparse, low-bias linear codes are locally testable (in particular sparse random codes are locally testable).
Kopparty ... more >>>

TR11-095 | 22nd June 2011
Christoph Behle, Andreas Krebs, Klaus-Joern Lange, Pierre McKenzie

#### Low uniform versions of NC1

Revisions: 1

In the setting known as DLOGTIME-uniformity,
the fundamental complexity classes
$AC^0\subset ACC^0\subseteq TC^0\subseteq NC^1$ have several
robust characterizations.
In this paper we refine uniformity further and examine the impact
of these refinements on $NC^1$ and its subclasses.
When applied to the logarithmic circuit depth characterization of $NC^1$,
some refinements leave ... more >>>

TR17-008 | 14th January 2017
Benny Applebaum, Naama Haramaty, Yuval Ishai, Eyal Kushilevitz, Vinod Vaikuntanathan

#### Low-Complexity Cryptographic Hash Functions

Cryptographic hash functions are efficiently computable functions that shrink a long input into a shorter output while achieving some of the useful security properties of a random function. The most common type of such hash functions is {\em collision resistant} hash functions (CRH), which prevent an efficient attacker from finding ... more >>>

TR22-082 | 27th May 2022
Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, João Ribeiro

#### Low-Degree Polynomials Extract from Local Sources

We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, ... more >>>

TR06-054 | 16th April 2006
Alex Samorodnitsky

#### Low-degree tests at large distances

We define tests of boolean functions which
distinguish between linear (or quadratic) polynomials, and functions
which are very far, in an appropriate sense, from these
polynomials. The tests have optimal or nearly optimal trade-offs
between soundness and the number of queries.

In particular, we show that functions with small ... more >>>

TR22-151 | 12th November 2022
Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, Bhargav Thankey

#### Low-depth arithmetic circuit lower bounds via shifted partials

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving ... more >>>

TR13-177 | 10th December 2013
Eric Allender, Nikhil Balaji, Samir Datta

#### Low-depth Uniform Threshold Circuits and the Bit-Complexity of Straight Line Programs

Revisions: 1

We present improved uniform TC$^0$ circuits for division, matrix powering, and related problems, where the improvement is in terms of majority depth'' (initially studied by Maciel and Therien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in ... more >>>

TR06-125 | 20th September 2006
Luis Antunes, Lance Fortnow, Alexandre Pinto, Andre Souto

#### Low-Depth Witnesses are Easy to Find

Antunes, Fortnow, van Melkebeek and Vinodchandran captured the
notion of non-random information by computational depth, the
difference between the polynomial-time-bounded Kolmogorov
complexity and traditional Kolmogorov complexity We show how to
find satisfying assignments for formulas that have at least one
assignment of logarithmic depth. The converse holds under a
standard ... more >>>

TR07-069 | 29th July 2007
Ronen Shaltiel, Chris Umans

#### Low-end uniform hardness vs. randomness tradeoffs for AM

In 1998, Impagliazzo and Wigderson proved a hardness vs. randomness tradeoff for BPP in the {\em uniform setting}, which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions by Trevisan and Vadhan (in a slightly weaker setting). In 2003, Gutfreund, Shaltiel and Ta-Shma proved ... more >>>

TR16-106 | 15th July 2016
Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

#### Low-error two-source extractors for polynomial min-entropy

Revisions: 1

We construct explicit two-source extractors for $n$ bit sources,
requiring $n^\alpha$ min-entropy and having error $2^{-n^\beta}$,
for some constants $0 < \alpha,\beta < 1$. Previously, constructions
for exponentially small error required either min-entropy
$0.49n$ \cite{Bou05} or three sources \cite{Li15}. The construction
combines somewhere-random condensers based on the Incidence
Theorem \cite{Zuc06,Li11}, ... more >>>

TR16-084 | 23rd May 2016
Shalev Ben-David

#### Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.1 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a ... more >>>

TR15-139 | 25th August 2015
Eli Ben-Sasson, Gal Maor

#### Lower bound for communication complexity with no public randomness

We give a self contained proof of a logarithmic lower bound on the communication complexity of any non redundant function, given that there is no access to shared randomness. This bound was first stated in Yao's seminal paper [STOC 1979], but no full proof appears in the literature.

Our proof ... more >>>

TR18-053 | 19th March 2018

#### Lower Bound for Non-Adaptive Estimate the Number of Defective Items

We prove that to estimate within a constant factor the number of defective items in a non-adaptive group testing algorithm we need at least $\tilde\Omega((\log n)(\log(1/\delta)))$ tests. This solves the open problem posed by Damaschke and Sheikh Muhammad.

more >>>

TR22-079 | 25th May 2022
Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, Rosie Zhao

#### Lower Bound Methods for Sign-rank and their Limitations

The sign-rank of a matrix $A$ with $\pm 1$ entries is the smallest rank of a real matrix with the same sign pattern as $A$. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is ... more >>>

TR16-153 | 28th September 2016
Christian Engels, Raghavendra Rao B V, Karteek Sreenivasaiah

#### Lower Bounds and Identity Testing for Projections of Power Symmetric Polynomials

Revisions: 3

The power symmetric polynomial on $n$ variables of degree $d$ is defined as
$p_d(x_1,\ldots, x_n) = x_{1}^{d}+\dots + x_{n}^{d}$. We study polynomials that are expressible as a sum of powers
of homogenous linear projections of power symmetric polynomials. These form a subclass of polynomials computed by
... more >>>

TR12-007 | 28th January 2012
Ruiwen Chen, Valentine Kabanets

#### Lower Bounds against Weakly Uniform Circuits

Revisions: 1

A family of Boolean circuits $\{C_n\}_{n\geq 0}$ is called \emph{$\gamma(n)$-weakly uniform} if
there is a polynomial-time algorithm for deciding the direct-connection language of every $C_n$,
given \emph{advice} of size $\gamma(n)$. This is a relaxation of the usual notion of uniformity, which allows one
to interpolate between complete uniformity (when $\gamma(n)=0$) ... more >>>

TR10-022 | 23rd February 2010
Vitaly Feldman, Homin Lee, Rocco Servedio

#### Lower Bounds and Hardness Amplification for Learning Shallow Monotone Formulas

Much work has been done on learning various classes of simple'' monotone functions under the uniform distribution. In this paper we give the first unconditional lower bounds for learning problems of this sort by showing that polynomial-time algorithms cannot learn constant-depth monotone Boolean formulas under the uniform distribution in the ... more >>>

TR17-077 | 30th April 2017
Guillaume Lagarde, Nutan Limaye, Srikanth Srinivasan

#### Lower Bounds and PIT for Non-Commutative Arithmetic circuits with Restricted Parse Trees

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $\mathbb{F}\langle x_1,\dots,x_N \rangle$, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse ... more >>>

TR08-006 | 18th January 2008
Ran Raz, Amir Yehudayoff

#### Lower Bounds and Separations for Constant Depth Multilinear Circuits

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth $d$ ... more >>>

TR98-036 | 11th June 1998
Vince Grolmusz, Gábor Tardos

#### Lower Bounds for (MOD p -- MOD m) Circuits

Modular gates are known to be immune for the random
restriction techniques of Ajtai; Furst, Saxe, Sipser; and Yao and
Hastad. We demonstrate here a random clustering technique which
overcomes this difficulty and is capable to prove generalizations of
several known modular circuit lower bounds of Barrington, Straubing,
Therien; Krause ... more >>>

TR19-128 | 24th September 2019
Anna Gal, Robert Robere

#### Lower Bounds for (Non-monotone) Comparator Circuits

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and ... more >>>

TR16-189 | 21st November 2016
Arnab Bhattacharyya, Sivakanth Gopi

#### Lower bounds for 2-query LCCs over large alphabet

A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates.
We show that any zero-error $2$-query locally correctable code $\mathcal{C}: \{0,1\}^k \to \Sigma^n$ that can correct a constant fraction of corrupted symbols must ... more >>>

TR16-107 | 17th July 2016
Nathanael Fijalkow

#### Lower Bounds for Alternating Online Space Complexity

Revisions: 1

The notion of online space complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm,
represented by a machine which scans the input exactly once from left to right.
In this paper, we study alternating machines as introduced by ... more >>>

TR14-026 | 27th February 2014
Jop Briet, Zeev Dvir, Guangda Hu, Shubhangi Saraf

#### Lower Bounds for Approximate LDCs

We study an approximate version of $q$-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A $q$-query $(\alpha,\delta)$-approximate LDC is a set $V$ of $n$ points in $\mathbb{R}^d$ so that, for each $i \in [d]$ there are $\Omega(\delta n)$ ... more >>>

TR17-185 | 28th November 2017
Makrand Sinha

#### Lower Bounds for Approximating the Matching Polytope

We prove that any extended formulation that approximates the matching polytope on $n$-vertex graphs up to a factor of $(1+\varepsilon)$ for any $\frac2n \le \varepsilon \le 1$ must have at least ${n}\choose{{\alpha}/{\varepsilon}}$ defining inequalities where $0<\alpha<1$ is an absolute constant. This is tight as exhibited by the $(1+\varepsilon)$ approximating linear ... more >>>

TR18-180 | 3rd November 2018
Nathanael Fijalkow, Guillaume Lagarde, Pierre Ohlmann, Olivier Serre

#### Lower bounds for arithmetic circuits via the Hankel matrix

We study the complexity of representing polynomials by arithmetic circuits in both the commutative and the non-commutative settings. Our approach goes through a precise understanding of the more restricted setting where multiplication is not associative, meaning that we distinguish $(xy)z$ from $x(yz)$.

Our first and main conceptual result is a ... more >>>

TR08-032 | 18th March 2008
Dmitriy Cherukhin

#### Lower Bounds for Boolean Circuits with Finite Depth and Arbitrary Gates

We consider bounded depth circuits over an arbitrary field $K$. If the field $K$ is finite, then we allow arbitrary gates $K^n\to K$. For instance, in the case of field $GF(2)$ we allow any Boolean gates. If the field $K$ is infinite, then we allow only polinomials.

For every fixed ... more >>>

TR03-004 | 24th December 2002

#### Lower Bounds for Bounded-Depth Frege Proofs via Buss-Pudlack Games

We present a simple proof of the bounded-depth Frege lower bounds of
Pitassi et. al. and Krajicek et. al. for the pigeonhole
principle. Our method uses the interpretation of proofs as two player
games given by Pudlak and Buss. Our lower bound is conceptually
simpler than previous ones, and relies ... more >>>

TR18-154 | 7th September 2018
Stasys Jukna, Andrzej Lingas

#### Lower Bounds for Circuits of Bounded Negation Width

We consider Boolean circuits over $\{\lor,\land,\neg\}$ with negations applied only to input variables. To measure the amount of negation'' in such circuits, we introduce the concept of their negation width.'' In particular, a circuit computing a monotone Boolean function $f(x_1,\ldots,x_n)$ has negation width $w$ if no nonzero term produced (purely ... more >>>

TR06-079 | 12th June 2006
Kristoffer Arnsfelt Hansen

#### Lower Bounds for Circuits with Few Modular Gates using Exponential Sums

We prove that any AC0 circuit augmented with {epsilon log^2 n}
MOD_m gates and with a MAJORITY gate at the output, require size
n^{Omega(log n)} to compute MOD_l, when l has a prime
factor not dividing m and epsilon is sufficiently small. We
also obtain ... more >>>

TR95-027 | 30th May 1995
Frederic Green

#### Lower Bounds for Circuits with Mod Gates and One Exact Threshold Gate

We say an integer polynomial $p$, on Boolean inputs, weakly
$m$-represents a Boolean function $f$ if $p$ is non-constant and is zero (mod
$m$), whenever $f$ is zero. In this paper we prove that if a polynomial
weakly $m$-represents the Mod$_q$ function on $n$ inputs, where $q$ and $m$
more >>>

TR15-012 | 24th January 2015
Mika Göös

#### Lower Bounds for Clique vs. Independent Set

We prove an $\omega(\log n)$ lower bound on the conondeterministic communication complexity of the Clique vs. Independent Set problem introduced by Yannakakis (STOC 1988, JCSS 1991). As a corollary, this implies superpolynomial lower bounds for the Alon--Saks--Seymour conjecture in graph theory. Our approach is to first exhibit a query complexity ... more >>>

TR15-185 | 24th November 2015
Arnab Bhattacharyya, Sivakanth Gopi

#### Lower bounds for constant query affine-invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is ... more >>>

TR18-186 | 6th November 2018
Emanuele Viola

#### Lower bounds for data structures with space close to maximum imply circuit lower bounds

Revisions: 1

Let $f:\{0,1\}^{n}\to\{0,1\}^{m}$ be a function computable by a circuit with
unbounded fan-in, arbitrary gates, $w$ wires and depth $d$. With
a very simple argument we show that the $m$-query problem corresponding
to $f$ has data structures with space $s=n+r$ and time $(w/r)^{d}$,
for any $r$. As a consequence, in the ... more >>>

TR13-100 | 15th July 2013
Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan

#### Lower bounds for depth $4$ formulas computing iterated matrix multiplication

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth $4$ arithmetic formula computing the product of $d$ generic matrices of size $n \times n$, IMM$_{n,d}$, has size $n^{\Omega(\sqrt{d})}$ as long as $d \leq n^{1/10}$. This improves the result of Nisan and Wigderson (Computational ... more >>>

TR13-068 | 3rd May 2013
Mrinal Kumar, Shubhangi Saraf

#### Lower Bounds for Depth 4 Homogenous Circuits with Bounded Top Fanin

We study the class of homogenous $\Sigma\Pi\Sigma\Pi(r)$ circuits, which are depth 4 homogenous circuits with top fanin bounded by $r$. We show that any homogenous $\Sigma\Pi\Sigma\Pi(r)$ circuit computing the permanent of an $n\times n$ matrix must have size at least $\exp\left(n^{\Omega(1/r)}\right)$.

In a recent result, Gupta, Kamath, Kayal and ... more >>>

TR14-089 | 16th July 2014
Neeraj Kayal, Chandan Saha

#### Lower Bounds for Depth Three Arithmetic Circuits with small bottom fanin

Revisions: 1

Shpilka and Wigderson (CCC 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field $\mathbb{F}$ of characteristic zero. We resolve this problem by proving a $N^{\Omega(\frac{d}{\tau})}$ lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin ... more >>>

TR10-120 | 27th July 2010
Noa Eidelstein, Alex Samorodnitsky

#### Lower bounds for designs in symmetric spaces

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube.

We prove lower bounds on designs in spaces with a large group ... more >>>

TR18-181 | 30th October 2018
Giuseppe Persiano, Kevin Yeo

#### Lower Bounds for Differentially Private RAMs

In this work, we study privacy-preserving storage primitives that are suitable for use in data analysis on outsourced databases within the differential privacy framework. The goal in differentially private data analysis is to disclose global properties of a group without compromising any individual’s privacy. Typically, differentially private adversaries only ever ... more >>>

TR13-116 | 29th August 2013
Albert Atserias, Moritz Müller, Sergi Oliva

#### Lower Bounds for DNF-Refutations of a Relativized Weak Pigeonhole Principle

The relativized weak pigeonhole principle states that if at least $2n$ out of $n^2$ pigeons fly into $n$ holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size $2^{(\log n)^{3/2-\epsilon}}$ for every $\epsilon > 0$ and every sufficiently ... more >>>

TR16-165 | 30th October 2016

#### Lower Bounds for Elimination via Weak Regularity

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let $f:\{0,1\}^n \to \{0,1\}$ be any boolean function. Alice and Bob get $k$ inputs ... more >>>

TR07-137 | 6th November 2007
Yijia Chen, Jörg Flum, Moritz Müller

#### Lower Bounds for Kernelizations

Among others, refining the methods of [Fortnow and Santhanam, ECCC Report TR07-096] we improve a result of this paper and show for any parameterized problem with a `linear weak OR'' and with NP-hard underlying classical problem that there is no polynomial reduction from the problem to itself that assigns to ... more >>>

TR19-007 | 17th January 2019

#### Lower Bounds for Linear Decision Lists

We demonstrate a lower bound technique for linear decision lists, which are decision lists where the queries are arbitrary linear threshold functions.
We use this technique to prove an explicit lower bound by showing that any linear decision list computing the function $MAJ \circ XOR$ requires size $2^{0.18 n}$. This ... more >>>

TR01-080 | 14th November 2001
Oded Goldreich, Howard Karloff, Leonard Schulman, Luca Trevisan

#### Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval

Revisions: 3

We prove that if a linear error correcting code
$\C:\{0,1\}^n\to\{0,1\}^m$ is such that a bit of the message can
be probabilistically reconstructed by looking at two entries of a
corrupted codeword, then $m = 2^{\Omega(n)}$. We also present
several extensions of this result.

We show a reduction from the ... more >>>

TR99-019 | 7th June 1999
Detlef Sieling

#### Lower Bounds for Linear Transformed OBDDs and FBDDs

Linear Transformed Ordered Binary Decision Diagrams (LTOBDDs) have
been suggested as a generalization of OBDDs for the representation and
manipulation of Boolean functions. Instead of variables as in the
case of OBDDs parities of variables may be tested at the nodes of an
LTOBDD. By this extension it is ... more >>>

TR03-057 | 21st July 2003
Scott Aaronson

#### Lower Bounds for Local Search by Quantum Arguments

The problem of finding a local minimum of a black-box function is central
for understanding local search as well as quantum adiabatic algorithms.
For functions on the Boolean hypercube {0,1}^n, we show a lower bound of
Omega(2^{n/4}/n) on the number of queries needed by a quantum computer to
solve this ... more >>>

TR05-053 | 4th May 2005
Paul Beame, Nathan Segerlind

#### Lower bounds for Lovasz-Schrijver systems and beyond follow from multiparty communication complexity

We prove that an \omega(log^3 n) lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an n^\omega(1) size lower bound for tree-like Lovasz-Schrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n^\Omega(1) lower bound for the (k+1)-party NOF communication complexity of set-disjointness ... more >>>

TR19-047 | 2nd April 2019
Mrinal Kumar, Ben Lee Volk

#### Lower Bounds for Matrix Factorization

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of ... more >>>

TR01-060 | 23rd August 2001
Amir Shpilka

#### Lower bounds for matrix product

We prove lower bounds on the number of product gates in bilinear
compute the product of two $n \times n$ matrices over finite fields.
In particular we obtain the following results:

1. We show that the number of product gates in any bilinear

TR00-029 | 30th April 2000
Ran Raz, Amir Shpilka

#### Lower Bounds for Matrix Product, in Bounded Depth Circuits with Arbitrary Gates

Revisions: 1

We prove super-linear lower bounds for the number of edges
in constant depth circuits with $n$ inputs and up to $n$ outputs.
Our lower bounds are proved for all types of constant depth
circuits, e.g., constant depth arithmetic circuits, constant depth
threshold circuits ... more >>>

TR20-166 | 9th November 2020

#### Lower Bounds for Monotone Arithmetic Circuits Via Communication Complexity

Revisions: 1

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first qualitative improvement to this classical result: we construct a family of polynomials $P_n$ in $n$ variables, each of its monomials has positive coefficient, such that $P_n$ can be computed ... more >>>

TR14-169 | 9th December 2014
Stasys Jukna

#### Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of evaluated to 1 by a given 0-1 input vector (with multiplicities ... more >>>

TR97-032 | 11th July 1997
Jan Johannsen

#### Lower Bounds for Monotone Real Circuit Depth and Formula Size and Tree-like Cutting Planes

Using a notion of real communication complexity recently
introduced by J. Krajicek, we prove a lower bound on the depth of
monotone real circuits and the size of monotone real formulas for
st-connectivity. This implies a super-polynomial speed-up of dag-like
over tree-like Cutting Planes proofs.

more >>>

TR95-001 | 1st January 1995
Amos Beimel, Anna Gal, Michael S. Paterson

#### Lower Bounds for Monotone Span Programs

The model of span programs is a linear algebraic model of
computation. Lower bounds for span programs imply lower bounds for
contact schemes, symmetric branching programs and for formula size.
Monotone span programs correspond also to linear secret-sharing schemes.
We present a new technique for proving lower bounds for ... more >>>

TR07-014 | 23rd January 2007
Amit Chakrabarti

#### Lower Bounds for Multi-Player Pointer Jumping

We consider the $k$-layer pointer jumping problem in the one-way
multi-party number-on-the-forehead communication model. In this problem,
the input is a layered directed graph with each vertex having outdegree
$1$, shared amongst $k$ players: Player~$i$ knows all layers {\em
except} the $i$th. The players must communicate, in the order
$1,2,\ldots,k$, ... more >>>

TR12-141 | 22nd October 2012
Dmitry Itsykson, Dmitry Sokolov

#### Lower bounds for myopic DPLL algorithms with a cut heuristic

The paper is devoted to lower bounds on the time complexity of DPLL algorithms that solve the satisfiability problem using a splitting strategy. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for resolution proofs. Lower bounds on satisfiable instances are ... more >>>

TR04-083 | 8th September 2004
Boaz Barak, Yehuda Lindell, Salil Vadhan

#### Lower Bounds for Non-Black-Box Zero Knowledge

We show new lower bounds and impossibility results for general (possibly <i>non-black-box</i>) zero-knowledge proofs and arguments. Our main results are that, under reasonable complexity assumptions:
<ol>
<li> There does not exist a two-round zero-knowledge <i>proof</i> system with perfect completeness for an NP-complete language. The previous impossibility result for two-round zero ... more >>>

TR00-042 | 21st June 2000
Lars Engebretsen

#### Lower Bounds for non-Boolean Constraint Satisfaction

Revisions: 1

We show that the k-CSP problem over a finite Abelian group G
cannot be approximated within |G|^{k-O(sqrt{k})}-epsilon, for
any constant epsilon>0, unless P=NP. This lower bound matches
well with the best known upper bound, |G|^{k-1}, of Serna,
Trevisan and Xhafa. The proof uses a combination of PCP
techniques---most notably a ... more >>>

TR15-022 | 9th February 2015
Nutan Limaye, Guillaume Malod, Srikanth Srinivasan

#### Lower bounds for non-commutative skew circuits

Revisions: 1

Nisan (STOC 1991) exhibited a polynomial which is computable by linear sized non-commutative circuits but requires exponential sized non-commutative algebraic branching programs. Nisan's hard polynomial is in fact computable by linear sized skew circuits (skew circuits are circuits where every multiplication gate has the property that all but one of ... more >>>

TR01-020 | 20th February 2001
N. S. Narayanaswamy, C.E. Veni Madhavan

#### Lower Bounds for OBDDs and Nisan's pseudorandom generator

We present a new boolean function for which any Ordered Binary
Decision Diagram (OBDD) computing it has an exponential number
of nodes. This boolean function is obtained from Nisan's
pseudorandom generator to derandomize space bounded randomized
algorithms. Though the relation between hardness and randomness for
computational models is well ... more >>>

TR19-055 | 9th April 2019
Kasper Green Larsen, Tal Malkin, Omri Weinstein, Kevin Yeo

#### Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search (ANN) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the ... more >>>

TR22-038 | 13th March 2022
Russell Impagliazzo, Sasank Mouli, Toniann Pitassi

#### Lower bounds for Polynomial Calculus with extension variables over finite fields

Revisions: 1

For every prime p > 0, every n > 0 and ? = O(logn), we show the existence
of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over F_p with M extension variables, each depending on at ... more >>>

TR13-083 | 7th June 2013
Miklos Ajtai

#### Lower Bounds for RAMs and Quantifier Elimination

For each natural number $d$ we consider a finite structure ${\bf M}_{d}$ whose universe is the set of all $0,1$-sequence of length $n=2^{d}$, each representing a natural number in the set $\lbrace 0,1,...,2^{n}-1\rbrace$ in binary form. The operations included in the structure are the four constants $0,1,2^{n}-1,n$, multiplication and addition ... more >>>

TR21-073 | 3rd June 2021
Emanuele Viola

#### Lower bounds for samplers and data structures via the cell-probe separator

Revisions: 2

Suppose that a distribution $S$ can be approximately sampled by an
efficient cell-probe algorithm. It is shown to be possible to restrict
the input to the algorithm so that its output distribution is still
not too far from $S$, and at the same time many output coordinates
are almost pairwise ... more >>>

TR15-073 | 25th April 2015
Neeraj Kayal, Chandan Saha

#### Lower Bounds for Sums of Products of Low arity Polynomials

We prove an exponential lower bound for expressing a polynomial as a sum of product of low arity polynomials. Specifically, we show that for the iterated matrix multiplication polynomial, $IMM_{d, n}$ (corresponding to the product of $d$ matrices of size $n \times n$ each), any expression of the form
more >>>

TR02-064 | 14th November 2002
Andrej Bogdanov, Luca Trevisan

#### Lower Bounds for Testing Bipartiteness in Dense Graphs

We consider the problem of testing bipartiteness in the adjacency
matrix model. The best known algorithm, due to Alon and Krivelevich,
distinguishes between bipartite graphs and graphs that are
$\epsilon$-far from bipartite using $O((1/\epsilon^2)polylog(1/epsilon)$
queries. We show that this is optimal for non-adaptive algorithms,
up to the ... more >>>

TR13-036 | 13th March 2013
Eric Blais, Sofya Raskhodnikova, Grigory Yaroslavtsev

#### Lower Bounds for Testing Properties of Functions on Hypergrid Domains

Revisions: 1

We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions $f : [n]^d \rightarrow \mathbb R$ on the hypergrid: monotonicity, convexity, and the Lipschitz property.
Our lower bounds also apply to the more restricted setting ... more >>>

TR09-066 | 13th August 2009
Arnab Bhattacharyya, Ning Xie

#### Lower Bounds for Testing Triangle-freeness in Boolean Functions

Let $f_{1},f_{2}, f_{3}:\mathbb{F}_{2}^{n} \to \{0,1\}$ be three Boolean functions.
We say a triple $(x,y,x+y)$ is a \emph{triangle} in the function-triple $(f_{1}, f_{2}, f_{3})$ if $f_{1}(x)=f_{2}(y)=f_{3}(x+y)=1$.
$(f_{1}, f_{2}, f_{3})$ is said to be \emph{triangle-free} if there is no triangle in the triple. The distance between a function-triple
and ... more >>>

TR14-150 | 10th November 2014
Justin Thaler

#### Lower Bounds for the Approximate Degree of Block-Composed Functions

Revisions: 3

We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials. Specifically, for any function $f$ on $N$ bits, define $F(x_1, \dots, x_M) = \text{OMB}(f(x_1), \dots, f(x_M))$ to be the function on $M \cdot N$ bits obtained by block-composing $f$ with a specific DNF known ... more >>>

TR14-139 | 31st October 2014
Hong Van Le

#### Lower bounds for the circuit size of partially homogeneous polynomials

Revisions: 1

In this paper
we associate to each multivariate polynomial $f$ that is homogeneous relative to a subset of its variables a series of polynomial families $P_\lambda (f)$ of $m$-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in $P_\lambda (f)$ is bounded from above ... more >>>

TR94-019 | 12th December 1994

#### Lower Bounds for the Computational Power of Networks of Spiking

We investigate the computational power of a formal model for networks of
spiking neurons. It is shown that simple operations on phase-differences
between spike-trains provide a very powerful computational tool that can
in principle be used to carry out highly complex computations on a small
network of spiking neurons. We ... more >>>

TR95-034 | 30th June 1995
Christoph Meinel, Stephan Waack

#### Lower Bounds for the Majority Communication Complexity of Various Graph Accessibility Problems

We investigate the probabilistic communication complexity
(more exactly, the majority communication complexity,)
of the graph accessibility problem GAP and
its counting versions MOD_k-GAP, k > 1.
Due to arguments concerning matrix variation ranks
and certain projection reductions, we prove
that, for any partition of the input variables,
more >>>

TR97-042 | 22nd September 1997
Russell Impagliazzo, Pavel Pudlak, Jiri Sgall

#### Lower Bounds for the Polynomial Calculus and the Groebner Basis Algorithm

Razborov~\cite{Razborov96} recently proved that polynomial
calculus proofs of the pigeonhole principle $PHP_n^m$ must have
degree at least $\ceiling{n/2}+1$ over any field. We present a
simplified proof of the same result. The main
idea of our proof is the same as in the original proof
of Razborov: we want to describe ... more >>>

TR03-068 | 30th July 2003
Matthias Homeister

#### Lower Bounds for the Sum of Graph--driven Read--Once Parity Branching Programs

We prove the first lower bound for restricted read-once parity branching
programs with unlimited parity nondeterminism where for each input the
variables may be tested according to several orderings.

Proving a superpolynomial lower bound for read-once parity branching
programs is still a challenging open problem. The following variant ... more >>>

TR18-094 | 2nd May 2018
Amit Levi, Erik Waingarten

#### Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

We introduce a new model for testing graph properties which we call the \emph{rejection sampling model}. We show that testing bipartiteness of $n$-nodes graphs using rejection sampling queries requires complexity $\widetilde{\Omega}(n^2)$. Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions ... more >>>

TR14-080 | 11th June 2014
Stasys Jukna

#### Lower Bounds for Tropical Circuits and Dynamic Programs

Revisions: 1

Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower ... more >>>

TR22-015 | 12th February 2022
Mika Göös, Stefan Kiefer, Weiqiang Yuan

#### Lower Bounds for Unambiguous Automata via Communication Complexity

We use results from communication complexity, both new and old ones, to prove lower bounds for unambiguous finite automata (UFAs). We show three results.

$\textbf{Complement:}$ There is a language $L$ recognised by an $n$-state UFA such that the complement language $\overline{L}$ requires NFAs with $n^{\tilde{\Omega}(\log n)}$ states. This improves on ... more >>>

TR10-085 | 20th May 2010
Eli Ben-Sasson, Jan Johannsen

#### Lower bounds for width-restricted clause learning on small width formulas

It has been observed empirically that clause learning does not significantly improve the performance of a SAT solver when restricted
to learning clauses of small width only. This experience is supported by lower bound theorems. It is shown that lower bounds on the runtime of width-restricted clause learning follow from ... more >>>

TR20-101 | 7th July 2020
Uma Girish, Ran Raz, Wei Zhan

#### Lower Bounds for XOR of Forrelations

The Forrelation problem, first introduced by Aaronson [AA10] and Aaronson and Ambainis [AA15], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [AA15]; the first separation between ... more >>>

TR20-089 | 8th June 2020
Dror Chawin, Iftach Haitner, Noam Mazor

#### Lower Bounds on the Time/Memory Tradeoff of Function Inversion

Revisions: 1

We study time/memory tradeoffs of function inversion: an algorithm, i.e., an inverter, equipped with an $s$-bit advice for a randomly chosen function $f\colon [n] \mapsto [n]$ and using $q$ oracle queries to $f$, tries to invert a randomly chosen output $y$ of $f$ (i.e., to find $x$ such that $f(x)=y$). ... more >>>

TR19-026 | 28th February 2019
Pavel Hrubes, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, Amir Yehudayoff

Revisions: 1

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets $S_1,\ldots,S_k \subset [n]$ is balancing if for every subset $X \subset \{1,2,\ldots,n\}$ of size $n/2$, there is an $i \in [k]$ so that $|S_i \cap X| = ... more >>> TR16-050 | 31st March 2016 Roei Tell #### Lower Bounds on Black-Box Reductions of Hitting to Density Estimation Revisions: 1 We consider the following problem. A deterministic algorithm tries to find a string in an unknown set$S\subseteq\{0,1\}^n$that is guaranteed to have large density (e.g.,$|S|\ge2^{n-1}$). However, the only information that the algorithm can obtain about$S$is estimates of the density of$S$in adaptively chosen subsets of ... more >>> TR12-038 | 6th April 2012 Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, David Xiao #### Lower bounds on information complexity via zero-communication protocols and applications We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck and prove that it lower bounds the information complexity of any function. Our relaxed partition ... more >>> TR12-108 | 4th September 2012 Arkadev Chattopadhyay, Rahul Santhanam #### Lower Bounds on Interactive Compressibility by Constant-Depth Circuits We formulate a new connection between instance compressibility \cite{Harnik-Naor10}), where the compressor uses circuits from a class$\C$, and correlation with circuits in$\C$. We use this connection to prove the first lower bounds on general probabilistic multi-round instance compression. We show that there is no probabilistic multi-round ... more >>> TR20-073 | 5th May 2020 Sam Buss, Dmitry Itsykson, Alexander Knop, Artur Riazanov, Dmitry Sokolov #### Lower Bounds on OBDD Proofs with Several Orders This paper is motivated by seeking lower bounds on OBDD($\land$, weakening, reordering) refutations, namely OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1-NBP($\land$) refutations based on read-once nondeterministic branching programs. These generalize OBDD($\land$, reordering) refutations. There are polynomial size 1-NBP($\land$) refutations of the pigeonhole principle, hence ... more >>> TR21-077 | 6th June 2021 Shir Peleg, Amir Shpilka, Ben Lee Volk #### Lower Bounds on Stabilizer Rank The stabilizer rank of a quantum state$\psi$is the minimal$r$such that$\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$for$c_j \in \mathbb{C}$and stabilizer states$\varphi_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the ... more >>> TR00-002 | 23rd December 1999 Michael Schmitt #### Lower Bounds on the Complexity of Approximating Continuous Functions by Sigmoidal Neural Networks We calculate lower bounds on the size of sigmoidal neural networks that approximate continuous functions. In particular, we show that for the approximation of polynomials the network size has to grow as$\Omega((\log k)^{1/4})$where$k$is the degree of the polynomials. This bound is ... more >>> TR04-120 | 22nd November 2004 Andris Ambainis, William Gasarch, Aravind Srinivasan, Andrey Utis #### Lower bounds on the Deterministic and Quantum Communication Complexity of HAM_n^a Alice and Bob want to know if two strings of length$n$are almost equal. That is, do they differ on at most$a$bits? Let$0\le a\le n-1$. We show that any deterministic protocol, as well as any error-free quantum protocol ($C^*$version), for this problem requires at ... more >>> TR00-022 | 2nd May 2000 Rosario Gennaro, Luca Trevisan #### Lower bounds on the efficiency of generic cryptographic constructions We present lower bounds on the efficiency of constructions for Pseudo-Random Generators (PRGs) and Universal One-Way Hash Functions (UOWHFs) based on black-box access to one-way permutations. Our lower bounds are tight as they match the efficiency of known constructions. A PRG (resp. UOWHF) construction based on black-box access is a ... more >>> TR11-016 | 7th February 2011 Sergei Artemenko, Ronen Shaltiel #### Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification Revisions: 1 Hardness amplification results show that for every function$f$there exists a function$Amp(f)$such that the following holds: if every circuit of size$s$computes$f$correctly on at most a$1-\delta$fraction of inputs, then every circuit of size$s'$computes$Amp(f)$correctly on at most a$1/2+\eps$... more >>> TR09-010 | 29th January 2009 Nikos Leonardos, Michael Saks #### Lower bounds on the randomized communication complexity of read-once functions We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond ... more >>> TR20-039 | 25th March 2020 Pranjal Dutta, Nitin Saxena, Thomas Thierauf #### Lower bounds on the sum of 25th-powers of univariates lead to complete derandomization of PIT We consider the univariate polynomial$f_d:=(x+1)^d$when represented as a sum of constant-powers of univariate polynomials. We define a natural measure for the model, the support-union, and conjecture that it is$\Omega(d)$for$f_d$. We show a stunning connection of the conjecture to the two main problems in algebraic ... more >>> TR17-190 | 6th November 2017 Anirbit Mukherjee, Amitabh Basu #### Lower bounds over Boolean inputs for deep neural networks with ReLU gates. Motivated by the resurgence of neural networks in being able to solve complex learning tasks we undertake a study of high depth networks using ReLU gates which implement the function$x \mapsto \max\{0,x\}$. We try to understand the role of depth in such neural networks by showing size lowerbounds against ... more >>> TR98-015 | 16th January 1998 Valentin E. Brimkov, Stefan S. Dantchev #### Lower Bounds, "Pseudopolynomial" and Approximation Algorithms for the Knapsack Problem with Real Coefficients In this paper we study the Boolean Knapsack problem (KP$_{{\bf R}}$)$a^Tx=1$,$x \in \{0,1\}^n$with real coefficients, in the framework of the Blum-Shub-Smale real number computational model \cite{BSS}. We obtain a new lower bound$\Omega \left( n\log n\right) \cdot f(1/a_{\min})\$ for the time
complexity ... more >>>

TR15-133 | 12th August 2015
Olaf Beyersdorff, Ilario Bonacina, Leroy Chew

#### Lower bounds: from circuits to QBF proof systems

A general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from ... more >>>

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