ECCC - Reports by year

Electronic Colloquium on Computational Complexity

REPORTS > 2009:

All reports in year 2009:

TR09-147 | 30th December 2009
Stephan Kreutzer

Algorithmic Meta-Theorems

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

TR09-146 | 29th December 2009
Dan Gutfreund, Akinori Kawachi

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

We show that if Arthur-Merlin protocols can be derandomized, then there is a Boolean function computable in deterministic exponential-time with access to an NP oracle, that cannot be computed by Boolean circuits of exponential size. More formally, if $\mathrm{prAM}\subseteq \mathrm{P}^{\mathrm{NP}}$ then there is a Boolean function in $\mathrm{E}^{\mathrm{NP}}$ that requires ... more >>>

TR09-145 | 20th December 2009
Shankar Kalyanaraman, Chris Umans

The Complexity of Rationalizing Network Formation

We study the complexity of {\em rationalizing} network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we ... more >>>

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

An Invariance Principle for Polytopes

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

TR09-143 | 22nd December 2009
Noam Livne

On the Construction of One-Way Functions from Average Case Hardness

In this paper we study the possibility of proving the existence of one-way functions based on average case hardness. It is well-known that if there exists a polynomial-time sampler that outputs instance-solution pairs such that the distribution on the instances is hard on average, then one-way functions exist. We study ... more >>>

TR09-142 | 17th December 2009
Aaron Potechin

Bounds on Monotone Switching Networks for Directed Connectivity

We prove that any monotone switching network solving directed connectivity on $N$ vertices must have size $N^{\Omega(\log N)}$ more >>>

TR09-141 | 19th December 2009
Anindya De, Omid Etesami, Luca Trevisan, Madhur Tulsiani

Improved Pseudorandom Generators for Depth 2 Circuits

We prove the existence of a $poly(n,m)$-time computable pseudorandom generator which ``$1/poly(n,m)$-fools'' DNFs with $n$ variables and $m$ terms, and has seed length $O(\log^2 nm \cdot \log\log nm)$. Previously, the best pseudorandom generator for depth-2 circuits had seed length $O(\log^3 nm)$, and was due to Bazzi (FOCS 2007). It follows ... more >>>

TR09-140 | 18th December 2009
Saugata Basu

A complex analogue of Toda's Theorem

Revisions: 1

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

TR09-139 | 17th December 2009
Mohammad Mahmoody, David Xiao

On the Power of Randomized Reductions and the Checkability of SAT

Revisions: 2

The closure of complexity classes is a elicate question and the answer varies depending on the type of reduction considered. The closure of most classes under many-to-one (Karp) reductions is clear, but the question becomes complicated when oracle (Cook) reductions are allowed, and even more so when the oracle reductions ... more >>>

TR09-138 | 14th December 2009
Gillat Kol, Ran Raz

Bounds on 2-Query Locally Testable Codes with Affine Tests

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

TR09-137 | 14th December 2009
Massimo Lauria

Random CNFs require spacious Polynomial Calculus refutations

Comments: 1

We study the space required by Polynomial Calculus refutations of random $k$-CNFs. We are interested in how many monomials one needs to keep in memory to carry on a refutation. More precisely we show that for $k \geq 4$ a refutation of a random $k$-CNF of $\Delta n$ clauses and ... more >>>

TR09-136 | 9th December 2009
Michael Burr, Felix Krahmer, Chee Yap

Continuous Amortization: A Non-Probabilistic Adaptive Analysis Technique

Let $f$ be a univariate polynomial with real coefficients, $f\in\RR[X]$. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the roots of $f$ in a given interval. In this paper, we consider subdivision algorithms based on purely numerical primitives such as function evaluation. ... more >>>

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

Monotone expanders - constructions and applications

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

TR09-134 | 10th December 2009
Zeev Dvir

On matrix rigidity and locally self-correctable codes

Revisions: 1

We describe a new approach for the problem of finding {\rm rigid} matrices, as posed by Valiant [Val77], by connecting it to the, seemingly unrelated, problem of proving lower bounds for locally self-correctable codes. This approach, if successful, could lead to a non-natural property (in the sense of Razborov and ... more >>>

TR09-133 | 9th December 2009
Anindya De, Thomas Vidick

Near-optimal extractors against quantum storage

We give near-optimal constructions of extractors secure against quantum bounded-storage adversaries. One instantiation gives the first such extractor to achieve an output length Theta(K-b), where K is the source's entropy and b the adversary's storage, depending linearly on the adversary's amount of storage, together with a poly-logarithmic seed length. Another ... more >>>

TR09-132 | 8th December 2009
Lior Malka

How to Achieve Perfect Simulation and a Complete Problem for Non-interactive Perfect Zero-Knowledge

Perfect zero-knowledge (PZK) proofs have been constructed in various settings and they are also interesting from a complexity theoretic perspective. Yet, virtually nothing is known about them. This is in contrast to statistical zero-knowledge proofs, where many general results have been proved using an array of tools, none of which ... more >>>

TR09-131 | 2nd December 2009
Stephan Kreutzer, Anuj Dawar

Parameterized Complexity of First-Order Logic

We show that if $\mathcal C$ is a class of graphs which is "nowhere dense" then first-order model-checking is fixed-parameter tractable on $\mathcal C$. As all graph classes which exclude a fixed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic ... more >>>

TR09-130 | 1st December 2009
Ryan O'Donnell, YI WU, Yuan Zhou

Optimal lower bounds for locality sensitive hashing (except when $q$ is tiny)

We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in $\{0,1\}^d$ under the Hamming distance. Recall that $\mathcal{H}$ is said to be an $(r, cr, p, q)$-sensitive hash family if all pairs $x,y \in \{0,1\}^d$ with dist$(x,y) \leq r$ have probability at least $p$ ... more >>>

TR09-129 | 30th November 2009
Boaz Barak, Moritz Hardt, Thomas Holenstein, David Steurer

Subsampling Semidefinite Programs and Max-Cut on the Sphere

We study the question of whether the value of mathematical programs such as linear and semidefinite programming hierarchies on a graph $G$, is preserved when taking a small random subgraph $G'$ of $G$. We show that the value of the Goemans-Williamson (1995) semidefinite program (SDP) for \maxcut of $G'$ is ... 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 >>>

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

Lossy Trapdoor Functions from Smooth Homomorphic Hash Proof Systems

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 (IND-CCA2) ... 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 codewords. ... more >>>

TR09-125 | 24th November 2009
David Steurer, Nisheeth Vishnoi

Connections Between Unique Games and Multicut

In this paper we demonstrate a close connection between UniqueGames and MultiCut. % In MultiCut, one is given a ``network graph'' and a ``demand graph'' on the same vertex set and the goal is to remove as few edges from the network graph as possible such that every two vertices ... more >>>

TR09-124 | 24th November 2009
Amit Kumar, Rajsekar Manokaran, Madhur Tulsiani, Nisheeth Vishnoi

On the Optimality of a Class of LP-based Algorithms

Revisions: 1

In this paper we will be concerned with a large class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, for NP-hard problems among them, one can write an LP relaxation and then round the solution. For instance, for Vertex Cover, one can obtain a 2-approximation ... more >>>

TR09-123 | 23rd November 2009
Hemanta Maji, Manoj Prabhakaran, Mike Rosulek

Cryptographic Complexity Classes and Computational Intractability Assumptions

Which computational intractability assumptions are inherent to cryptography? We present a broad framework to pose and investigate this question. We first aim to understand the “cryptographic complexity” of various tasks, independent of any computational assumptions. In our framework the cryptographic tasks are modeled as multi- party computation functionalities. We consider ... more >>>

TR09-122 | 23rd November 2009
Vikraman Arvind, Srikanth Srinivasan

Circuit Lower Bounds, Help Functions, and the Remote Point Problem

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving ... more >>>

TR09-121 | 22nd November 2009
Zohar Karnin, Yuval Rabani, Amir Shpilka

Explicit Dimension Reduction and Its Applications

We construct a small set of explicit linear transformations mapping $R^n$ to $R^{O(\log n)}$, such that the $L_2$ norm of any vector in $R^n$ is distorted by at most $1\pm o(1)$ in at least a fraction of $1 - o(1)$ of the transformations in the set. Albeit the tradeoff between ... more >>>

TR09-120 | 18th November 2009
Charanjit Jutla

Almost Optimal Bounds for Direct Product Threshold Theorem

Revisions: 1

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

TR09-119 | 17th November 2009
Frederic Magniez, Claire Mathieu, Ashwin Nayak

Recognizing well-parenthesized expressions in the streaming model

Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis. We present a one-pass randomized streaming algorithm ... more >>>

TR09-118 | 18th November 2009
Shachar Lovett, Ido Ben-Eliezer, Ariel Yadin

Title: Polynomial Threshold Functions: Structure, Approximation and Pseudorandomness

Revisions: 1

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We ... more >>>

TR09-117 | 18th November 2009
Ilias Diakonikolas, Daniel Kane, Jelani Nelson

Bounded Independence Fools Degree-2 Threshold Functions

Revisions: 1

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of ... more >>>

TR09-116 | 15th November 2009
Zohar Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich

Deterministic identity testing of depth 4 multilinear circuits with bounded top fan-in

We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-$4$ multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-$4$ circuits with a plus gate at the top (also known as $\Spsp$ circuits) and has a running time of $\exp(\poly(\log(n),\log(s),k))$ where $n$ is ... 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 for ... more >>>

TR09-114 | 13th November 2009
Emanuele Viola

Are all distributions easy?

Complexity theory typically studies the complexity of computing a function $h(x) : \{0,1\}^n \to \{0,1\}^m$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits. Our main results are: \begin{itemize} \item There are explicit $AC^0$ circuits of ... more >>>

TR09-113 | 9th November 2009
Anindya De, Luca Trevisan, Madhur Tulsiani

Non-uniform attacks against one-way functions and PRGs

We study the power of non-uniform attacks against one-way functions and pseudorandom generators. Fiat and Naor show that for every function $f: [N]\to [N]$ there is an algorithm that inverts $f$ everywhere using (ignoring lower order factors) time, space and advice at most $N^{3/4}$. We show that an algorithm using ... more >>>

TR09-112 | 3rd November 2009
Davide Bilò, Luciano Gualà, Guido Proietti

Hardness of an Asymmetric 2-player Stackelberg Network Pricing Game

Consider a communication network represented by a directed graph $G=(V,E)$ of $n$ nodes and $m$ edges. Assume that edges in $E$ are partitioned into two sets: a set $C$ of edges with a fixed non-negative real cost, and a set $P$ of edges whose \emph{price} should instead be set by ... more >>>

TR09-111 | 5th November 2009
Walid Gomaa

Model-Theoretic Characterization of Complexity Classes

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

TR09-110 | 5th November 2009
Scott Aaronson, Andris Ambainis

The Need for Structure in Quantum Speedups

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that ... more >>>

TR09-109 | 3rd November 2009
Kai-Min Chung, Feng-Hao Liu

Tight Parallel Repetition Theorems for Public-coin Arguments

Following Hastad, Pass, Pietrzak, and Wikstrom (2008), we study parallel repetition theorems for public-coin interactive arguments and their generalization. We obtain the following results: 1. We show that the reduction of Hastad et al. actually gives a tight direct product theorem for public-coin interactive arguments. That is, $n$-fold parallel repetition ... more >>>

TR09-108 | 31st October 2009
Chongwon Cho, Chen-Kuei Lee, Rafail Ostrovsky

Equivalence of Uniform Key Agreement and Composition Insecurity

It is well known that proving the security of a key agreement protocol (even in a special case where the protocol transcript looks random to an outside observer) is at least as difficult as proving $P \not = NP$. Another (seemingly unrelated) statement in cryptography is the existence of two ... more >>>

TR09-107 | 28th October 2009
Kevin Dick, Chris Umans

Improved inapproximability factors for some $\Sigma_2^p$ minimization problems

We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are $\Sigma_2^p$-hard to approximate to within factors of $n^{1/3-\epsilon}$ and $n^{1/2-\epsilon}$ (where the previous results achieved $n^{1/4 - \epsilon}$), ... more >>>

TR09-106 | 28th October 2009
Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar, Jayalal Sarma

Using Elimination Theory to construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found ... more >>>

TR09-105 | 27th October 2009
Vikraman Arvind, Srikanth Srinivasan

The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets

Using $\epsilon$-bias spaces over F_2 , we show that the Remote Point Problem (RPP), introduced by Alon et al [APY09], has an $NC^2$ algorithm (achieving the same parameters as [APY09]). We study a generalization of the Remote Point Problem to groups: we replace F_n by G^n for an arbitrary fixed ... more >>>

TR09-104 | 26th October 2009
Scott Aaronson

BQP and the Polynomial Hierarchy

The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there ... more >>>

TR09-103 | 26th October 2009
Vikraman Arvind, Srikanth Srinivasan

On the Hardness of the Noncommutative Determinant

In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: ... more >>>

TR09-102 | 21st October 2009
Andrew Drucker, Ronald de Wolf

Quantum Proofs for Classical Theorems

Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (non-quantum) areas. In this paper we survey these results and the quantum toolbox they use. more >>>

TR09-101 | 20th October 2009
Nitin Saxena

Progress on Polynomial Identity Testing

Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem ... more >>>

TR09-100 | 16th October 2009
Jakob Nordström, Alexander Razborov

On Minimal Unsatisfiability and Time-Space Trade-offs for $k$-DNF Resolution

In the context of proving lower bounds on proof space in $k$-DNF resolution, [Ben-Sasson and Nordström 2009] introduced the concept of minimally unsatisfiable sets of $k$-DNF formulas and proved that a minimally unsatisfiable $k$-DNF set with $m$ formulas can have at most $O((mk)^{k+1})$ variables. They also gave an example of ... more >>>

TR09-099 | 16th October 2009
Venkatesan Guruswami, Ali Kemal Sinop

Improved Inapproximability Results for Maximum k-Colorable Subgraph

Revisions: 1

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a $k$-colorable graph with $k$ colors so that a maximum fraction of edges are properly colored (i.e., their endpoints receive different colors). A random $k$-coloring properly colors an expected fraction ... more >>>

TR09-098 | 9th October 2009
Alexander A. Sherstov

The intersection of two halfspaces has high threshold degree

Revisions: 1

The threshold degree of a Boolean function $f\colon\{0,1\}\to\{-1,+1\}$ is the least degree of a real polynomial $p$ such $f(x)\equiv\mathrm{sgn}\; p(x).$ We construct two halfspaces on $\{0,1\}^n$ whose intersection has threshold degree $\Theta(\sqrt n),$ an exponential improvement on previous lower bounds. This solves an open problem due to Klivans (2002) and ... more >>>

TR09-097 | 2nd September 2009
Rakesh Mohanty, N. S. Narayanaswamy

Online Algorithms for Self-Organizing Sequential Search - A Survey

The main objective of this survey is to present the important theoretical and experimental results contributed till date in the area of online algorithms for the self organizing sequential search problem, also popularly known as the List Update Problem(LUP) in a chronological way. The survey includes competitiveness results of deterministic ... more >>>

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

A Graph Spectral Approach for Computing Approximate Nash Equilibria

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

TR09-095 | 25th September 2009
Swapnoneel Roy

STRIP EXCHANGING IS HARD

Revisions: 1

The sorting by strip exchanges (aka strip exchanging) problem is motivated by applications in optical character recognition and genome rearrangement analysis. While a couple of approximation algorithms have been designed for the problem, nothing has been known about its computational hardness till date. In this work, we show that the ... more >>>

TR09-094 | 7th October 2009
Bireswar Das, Jacobo Toran, Fabian Wagner

Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time \cite{Bo90},\cite{YBFT}. We give restricted space algorithms for these problems proving the following results: Isomorphism for bounded tree distance width graphs is in L and thus complete for ... more >>>

TR09-093 | 8th October 2009
Vikraman Arvind, Bireswar Das, Johannes Köbler, Seinosuke Toda

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism} which has running time $b!2^{O(b)}N^{O(1)}$, where the parameter $b$ is the maximum size of the color classes of the given hypergraphs and $N$ is the input size. We also describe fpt algorithms for certain permutation group problems that ... more >>>

TR09-092 | 8th October 2009
Olaf Beyersdorff, Johannes Köbler, Sebastian Müller

Proof Systems that Take Advice

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow (JSL 79), where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajicek (JSL 07) have ... more >>>

TR09-091 | 23rd September 2009
Thanh Minh Hoang

On the Matching Problem for Special Graph Classes

Revisions: 1

In the present paper we show some new complexity bounds for the matching problem for special graph classes. We show that for graphs with a polynomially bounded number of nice cycles, the decision perfect matching problem is in $SPL$, it is hard for $FewL$, and the construction perfect matching problem ... more >>>

TR09-090 | 6th October 2009
Russell Impagliazzo, Valentine Kabanets, Avi Wigderson

New Direct-Product Testers and 2-Query PCPs

The “direct product code” of a function f gives its values on all k-tuples (f(x1), . . . , f(xk)). This basic construct underlies “hardness amplification” in cryptography, circuit complexity and PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent result by Dinur and ... more >>>

TR09-089 | 26th September 2009
Guy Rothblum, Salil Vadhan

Are PCPs Inherent in Efficient Arguments?

Starting with Kilian (STOC `92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collision-resistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC `07) raised the question of whether PCPs ... more >>>

TR09-088 | 29th September 2009
Shachar Lovett, Yoav Tzur

Explicit lower bound for fooling polynomials by the sum of small-bias generators

Recently, Viola (CCC'08) showed that the sum of $d$ small-biased distributions fools degree-$d$ polynomial tests; that is, every polynomial expression of degree at most $d$ in the bits of the sum has distribution very close to that induced by this expression evaluated on uniformly selected random bits. We show that ... more >>>

TR09-087 | 1st October 2009
Olga Tveretina, Carsten Sinz, Hans Zantema

Ordered Binary Decision Diagrams, Pigeonhole Formulas and Beyond

Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential size and that limited OBDD derivations cannot simulate resolution polynomially. Here we show that any arbitrary OBDD Apply refutation of the pigeonhole formula has an exponential size: we prove that the size of one ... more >>>

TR09-086 | 2nd October 2009
Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, David Zuckerman

Optimal testing of Reed-Muller codes

We consider the problem of testing if a given function $f : \F_2^n \rightarrow \F_2$ is close to any degree $d$ polynomial in $n$ variables, also known as the Reed-Muller testing problem. Alon et al.~\cite{AKKLR} proposed and analyzed a natural $2^{d+1}$-query test for this property and showed that it accepts ... more >>>

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

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

Revisions: 1

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

TR09-084 | 24th September 2009
Arkadev Chattopadhyay, Avi Wigderson

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 >>>

TR09-083 | 24th September 2009
Dana Ron, Mira Gonen, Yuval Shavitt

Counting Stars and Other Small Subgraphs in Sublinear Time

Detecting and counting the number of copies of certain subgraphs (also known as {\em network motifs\/} or {\em graphlets\/}), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of ... more >>>

TR09-082 | 20th September 2009
Vijayaraghavan T.C.

Characterization of ModL using Prime Modulus.

The complexity class ModL was defined by Arvind and Vijayaraghavan in [AV04] (more precisely in Definition 1.4.1, Vij08],[Definition 3.1, AV]). In this paper, under the assumption that NL =UL, we show that for every language $L\in ModL$ there exists a function $f\in \sharpL$ and a function $g\in FL$ such that ... more >>>

TR09-081 | 27th August 2009
Olaf Beyersdorff, Zenon Sadowski

Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes

In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT and concrete promise classes C (such as NP ... more >>>

TR09-080 | 19th September 2009
Elad Haramaty, Amir Shpilka

On the Structure of Cubic and Quartic Polynomials

Revisions: 1

In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias ... more >>>

TR09-079 | 21st September 2009
Victor Chen, Elena Grigorescu, Ronald de Wolf

Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation

Revisions: 1

We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers most queries correctly with high probability and for the remaining queries, the decoder with high probability either answers correctly or declares ``don't know.'' Furthermore, if there is no ... more >>>

TR09-078 | 16th September 2009
Falk Unger

A Probabilistic Inequality with Applications to Threshold Direct-product Theorems

We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments. This inequality allows us to simplify and strengthen several known direct-product ... more >>>

TR09-077 | 16th September 2009
Zeev Dvir

From Randomness Extraction to Rotating Needles

The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this problem ... more >>>

TR09-076 | 19th August 2009
Christian Glaßer, Christian Reitwießner, Maximilian Witek

Improved and Derandomized Approximations for Two-Criteria Metric Traveling Salesman

Revisions: 1

We improve and derandomize the best known approximation algorithm for the two-criteria metric traveling salesman problem (2-TSP). More precisely, we construct a deterministic 2-approximation which answers an open question by Manthey. Moreover, we show that 2-TSP is randomized $(3/2+\epsilon ,2)$-approximable, and we give the first randomized approximations for the two-criteria ... more >>>

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

A Theory of Goal-Oriented Communication

Comments: 1

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

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

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

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

TR09-073 | 6th September 2009
Vikraman Arvind, Pushkar Joglekar, Srikanth Srinivasan

On Lower Bounds for Constant Width Arithmetic Circuits

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of ... more >>>

TR09-072 | 3rd September 2009
Paul Beame, Trinh Huynh

Hardness Amplification in Proof Complexity

Revisions: 2 , Comments: 1

We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most $k$ (known as ${\rm Th}(k)$ proofs). Such systems include: Lovasz-Schrijver ... more >>>

TR09-071 | 1st September 2009
John Hitchcock, A. Pavan, N. V. Vinodchandran

Kolmogorov Complexity in Randomness Extraction

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction ... more >>>

TR09-070 | 1st September 2009
Andrej Bogdanov, Zeev Dvir, Elad Verbin, Amir Yehudayoff

Pseudorandomness for Width 2 Branching Programs

Bogdanov and Viola (FOCS 2007) constructed a pseudorandom generator that fools degree $k$ polynomials over $\F_2$ for an arbitrary constant $k$. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read $k$ bits of inputs at a time. This ... more >>>

TR09-069 | 2nd September 2009
Parikshit Gopalan

A note on Efremenko's Locally Decodable Codes

Revisions: 1

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

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 >>>

TR09-067 | 18th August 2009
Hanna Mazzawi, Nader Bshouty

On Parity Check $(0,1)$-Matrix over $Z_p$

We prove that for every prime $p$ there exists a $(0,1)$-matrix $M$ of size $t_p(n,m)\times n$ where $$t_p(n,m)=O\left(m+\frac{m\log \frac{n}{m}}{\log \min({m,p})}\right)$$ such that every $m$ columns of $M$ are linearly independent over $\Z_p$, the field of integers modulo $p$ (and therefore over any field of characteristic $p$ and over the real ... 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 triangle-freeness is the ... more >>>

TR09-065 | 31st July 2009
Panagiotis Voulgaris, Daniele Micciancio

Faster exponential time algorithms for the shortest vector problem

We present new faster algorithms for the exact solution of the shortest vector problem in arbitrary lattices. Our main result shows that the shortest vector in any $n$-dimensional lattice can be found in time $2^{3.199 n}$ and space $2^{1.325 n}$. This improves the best previously known algorithm by Ajtai, Kumar ... more >>>

TR09-064 | 3rd August 2009
Harry Buhrman, Lance Fortnow, Rahul Santhanam

Unconditional Lower Bounds against Advice

We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: \begin{enumerate} \item For any constant $c$, $\NEXP \not \subseteq \P^{\NP[n^c]}/n^c$ \item For any constant $c$, $\MAEXP \not \subseteq \MA/n^c$ \item $\BPEXP \not \subseteq \BPP/n^{o(1)}$ \end{enumerate} It was previously unknown even whether $\NEXP \subseteq ... more >>>

TR09-063 | 29th July 2009
Matt DeVos, Ariel Gabizon

Simple Affine Extractors using Dimension Expansion

Revisions: 2

Let $\F$ be the field of $q$ elements. An \emph{\afsext{n}{k}} is a mapping $D:\F^n\ar\B$ such that for any $k$-dimensional affine subspace $X\subseteq \F^n$, $D(x)$ is an almost unbiased bit when $x$ is chosen uniformly from $X$. Loosely speaking, the problem of explicitly constructing affine extractors gets harder as $q$ gets ... 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 that ... more >>>

TR09-061 | 2nd July 2009
Konstantinos Georgiou, Avner Magen, Madhur Tulsiani

Optimal Sherali-Adams Gaps from Pairwise Independence

This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX k-CSP(P)). We show that if the set of assignments accepted by $P$ contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on $n$ variables cannot be approximated ... 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 >>>

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

A NEW LINE OF ATTACK ON THE DICHOTOMY CONJECTURE

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

TR09-058 | 4th July 2009
Gábor Ivanyos, Marek Karpinski, Nitin Saxena

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundamental problems in computational complexity ... more >>>

TR09-057 | 23rd June 2009
Yonatan Bilu, Nathan Linial

Are stable instances easy?

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly ... more >>>

TR09-056 | 20th June 2009
Hunter Monroe

Speedup for Natural Problems and coNP?=NP

Revisions: 1

Informally, a language L has speedup if, for any Turing machine for L, there exists one that is better. Blum showed that there are computable languages that have almost-everywhere speedup. These languages were unnatural in that they were constructed for the sole purpose of having such speedup. We identify a ... more >>>

TR09-055 | 10th June 2009
Venkatesan Chakaravarthy, Sambuddha Roy

Arthur and Merlin as Oracles

We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) the Arthur-Merlin class. Our main results are the following: (i) $BPP^{NP}_{||} \subseteq P^{prAM}_{||}$; (ii) $S_2^p \subseteq P^{prAM}$. In addition to providing new upperbounds for the classes $S_2^p$ and $BPP^{NP}_{||}$, these results are interesting ... more >>>

TR09-054 | 7th June 2009
Emanuele Viola, Emanuele Viola

Cell-Probe Lower Bounds for Prefix Sums

We prove that to store n bits x so that each prefix-sum query Sum(i) := sum_{k < i} x_k can be answered by non-adaptively probing q cells of log n bits, one needs memory > n + n/log^{O(q)} n. Our bound matches a recent upper bound of n + n/log^{Omega(q)} ... more >>>

TR09-053 | 20th May 2009
Johannes Köbler, Sebastian Kuhnert

The Isomorphism Problem for k-Trees is Complete for Logspace

We show that k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for ... more >>>

TR09-052 | 2nd May 2009
Fabian Wagner, Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf

Planar Graph Isomorphism is in Log-space

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important ... more >>>

TR09-051 | 2nd July 2009
Eric Allender, Michal Koucký, Detlef Ronneburger, Sambuddha Roy

The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory

We continue an investigation into resource-bounded Kolmogorov complexity \cite{abkmr}, which highlights the close connections between circuit complexity and Levin's time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced ... 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 >>>

TR09-049 | 5th May 2009
Derrick Stolee, Derrick Stolee, Chris Bourke, N. V. Vinodchandran

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

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

TR09-048 | 29th May 2009
Parikshit Gopalan, Shachar Lovett, Amir Shpilka

On the Complexity of Boolean Functions in Different Characteristics

Every Boolean function on $n$ variables can be expressed as a unique multivariate polynomial modulo $p$ for every prime $p$. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree ... more >>>

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

A Space Hierarchy for k-DNF Resolution

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

TR09-046 | 9th May 2009
Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff

Transitive-Closure Spanners of the Hypercube and the Hypergrid

Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. Transitive-closure spanners were introduced in \cite{tc-spanners-soda} as a common abstraction ... more >>>

TR09-045 | 20th May 2009
Iftach Haitner, Omer Reingold, Salil Vadhan, Hoeteck Wee

Inaccessible Entropy

We put forth a new computational notion of entropy, which measures the (in)feasibility of sampling high entropy strings that are consistent with a given protocol. Specifically, we say that the i'th round of a protocol (A, B) has _accessible entropy_ at most k, if no polynomial-time strategy A^* can generate ... more >>>

TR09-044 | 6th May 2009
Boaz Barak, Mark Braverman, Xi Chen, Anup Rao

Direct Sums in Randomized Communication Complexity

Does computing n copies of a function require n times the computational effort? In this work, we give the first non-trivial answer to this question for the model of randomized communication complexity. We show that: 1. Computing n copies of a function requires sqrt{n} times the communication. 2. For average ... more >>>

TR09-043 | 18th May 2009
Elena Grigorescu, Tali Kaufman, Madhu Sudan

Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear error-correcting codes that have the ``single local orbit'' property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every ``sparse'' binary code whose coordinates ... more >>>

TR09-042 | 5th May 2009
Irit Dinur, Prahladh Harsha

Composition of low-error 2-query PCPs using decodable PCPs

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored ... more >>>

TR09-041 | 9th April 2009
Shiva Kintali, Laura J Poplawski, Rajmohan Rajaraman, Ravi Sundaram, Shang-Hua Teng

Reducibility Among Fractional Stability Problems

"As has often been the case with NP-completeness proofs, PPAD-completeness proofs will be eventually refined to cover simpler and more realistic looking classes of games. And then researchers will strive to identify even simpler classes." --Papadimitriou (chapter 2 of Algorithmic Game Theory book) In a landmark paper, Papadimitriou introduced a ... more >>>

TR09-040 | 20th April 2009
Pavel Hrubes, Stasys Jukna, Alexander Kulikov, Pavel Pudlak

On convex complexity measures

Khrapchenko's classical lower bound $n^2$ on the formula size of the parity function~$f$ can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle $f^{-1}(0)\times f^{-1}(1)$. Trying to generalize this approach we arrived at the concept of \emph{convex measures}. We prove the negative result that convex measures ... more >>>

TR09-039 | 6th April 2009
Matei David, Periklis Papakonstantinou, Anastasios Sidiropoulos

Polynomial Time with Restricted Use of Randomness

We define a hierarchy of complexity classes that lie between P and RP, yielding a new way of quantifying partial progress towards the derandomization of RP. A standard approach in derandomization is to reduce the number of random bits an algorithm uses. We instead focus on a model of computation ... more >>>

TR09-038 | 14th April 2009
Michael Alekhnovich, Allan Borodin, Joshua Buresh-Oppenheim, Russell Impagliazzo, Avner Magen

Toward a Model for Backtracking and Dynamic Programming

We propose a model called priority branching trees (pBT ) for backtracking and dynamic programming algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the ... more >>>

TR09-037 | 10th April 2009
Parikshit Gopalan

A Fourier-analytic approach to Reed-Muller decoding

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

TR09-036 | 14th April 2009
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

The Power of Depth 2 Circuits over Algebras

We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (Pi-Sigma) arithmetic circuits over U_2(F), the algebra of ... more >>>

TR09-035 | 26th March 2009
Nicola Galesi, Massimo Lauria

On the Automatizability of Polynomial Calculus

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Computing, 38(4), 2008). more >>>

TR09-034 | 25th March 2009
Eli Ben-Sasson, Jakob Nordström

Understanding Space in Resolution: Optimal Lower Bounds and Exponential Trade-offs

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. Understanding time and memory consumption, and how they are related to one another, is therefore a question of considerable practical importance. In the field of ... more >>>

TR09-033 | 16th April 2009
Phokion G. Kolaitis, Swastik Kopparty

Random Graphs and the Parity Quantifier

The classical zero-one law for first-order logic on random graphs says that for every first-order property $\varphi$ in the theory of graphs and every $p \in (0,1)$, the probability that the random graph $G(n, p)$ satisfies $\varphi$ approaches either $0$ or $1$ as $n$ approaches infinity. It is well known ... more >>>

TR09-032 | 16th April 2009
Neeraj Kayal, Shubhangi Saraf

Blackbox Polynomial Identity Testing for Depth 3 Circuits

We study depth three arithmetic circuits with bounded top fanin. We give the first deterministic polynomial time blackbox identity test for depth three circuits with bounded top fanin over the field of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is ... more >>>

TR09-031 | 6th April 2009
Zvika Brakerski, Oded Goldreich

From absolute distinguishability to positive distinguishability

We study methods of converting algorithms that distinguish pairs of distributions with a gap that has an absolute value that is noticeable into corresponding algorithms in which the gap is always positive. Our focus is on designing algorithms that, in addition to the tested string, obtain a fixed number of ... more >>>

TR09-030 | 5th April 2009
Shachar Lovett

The density of weights of Generalized Reed-Muller codes

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We ... more >>>

TR09-029 | 3rd April 2009
Fabian Wagner, Thomas Thierauf

Reachability in K_{3,3}-free Graphs and K_5-free Graphs is in Unambiguous Log-Space

Revisions: 1

We show that the reachability problem for directed graphs that are either K_{3,3}-free or K_5-free is in unambiguous log-space, UL \cap coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL \cap coUL. Our algorithm decomposes the graphs ... more >>>

TR09-028 | 2nd April 2009
Oded Goldreich

A Candidate Counterexample to the Easy Cylinders Conjecture

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

TR09-027 | 2nd April 2009
Iftach Haitner

A Parallel Repetition Theorem for Any Interactive Argument

Revisions: 1

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

TR09-026 | 17th February 2009
Vikraman Arvind, Pushkar Joglekar

Arithmetic Circuit Size, Identity Testing, and Finite Automata

Let $\F\{x_1,x_2,\cdots,x_n\}$ be the noncommutative polynomial ring over a field $\F$, where the $x_i$'s are free noncommuting formal variables. Given a finite automaton $\A$ with the $x_i$'s as alphabet, we can define polynomials $\f( mod A)$ and $\f(div A)$ obtained by natural operations that we call \emph{intersecting} and \emph{quotienting} the ... more >>>

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

A New Look at Some Classical Results in Computational Complexity

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

TR09-024 | 26th February 2009
Raghav Kulkarni

On the Power of Isolation in Planar Structures

The purpose of this paper is to study the deterministic {\em isolation} for certain structures in directed and undirected planar graphs. The motivation behind this work is a recent development on this topic. For example, \cite{btv07} isolate a directed path in planar graphs and \cite{dkr08} isolate a perfect matching in ... more >>>

TR09-023 | 12th March 2009
Akinori Kawachi, Osamu Watanabe

Strong Hardness Preserving Reduction from a P-Samplable Distribution to the Uniform Distribution for NP-Search Problems

Impagliazzo and Levin demonstrated [IL90] that the average-case hardness of any NP-search problem under any P-samplable distribution implies that of another NP-search problem under the uniform distribution. For this they developed a way to define a reduction from an NP-search problem F with ``mild hardness'' under any P-samplable distribution H; ... more >>>

TR09-022 | 16th February 2009
Jack H. Lutz, Elvira Mayordomo

Inseparability and Strong Hypotheses for Disjoint NP Pairs

Revisions: 1

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k)-inseparable. We also relate ... more >>>

TR09-021 | 2nd March 2009
Konstantin Makarychev, Yury Makarychev

How to Play Unique Games on Expanders

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a (1 - epsilon)-satisfiable instance of Unique Games with the constraint graph G, our algorithm finds an assignments satisfying at least a (1 - C ... more >>>

TR09-020 | 2nd March 2009
Venkatesan Guruswami, Prasad Raghavendra

Hardness of Solving Sparse Overdetermined Linear Systems: A 3-Query PCP over Integers.

A classic result due to Hastad established that for every constant \eps > 0, given an overdetermined system of linear equations over a finite field \F_q where each equation depends on exactly 3 variables and at least a fraction (1-\eps) of the equations can be satisfied, it is NP-hard to ... more >>>

TR09-019 | 10th March 2009
Agrawal Manindra, Osamu Watanabe

One-Way Functions and the Isomorphism Conjecture

We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomial-time many-one reductions are P-isomorphic to each other. From previous research it has been widely believed that all NP-complete sets are reducible each other by one-to-one and length-increasing polynomial-time reductions, but ... more >>>

TR09-018 | 8th March 2009
Yoav Tzur

$GF(2^n)$-Linear Tests versus $GF(2)$-Linear Tests

Comments: 1

A small-biased distribution of bit sequences is defined as one withstanding $GF(2)$-linear tests for randomness, which are linear combinations of the bits themselves. We consider linear combinations over larger fields, specifically, $GF(2^n)$ for $n$ that divides the length of the bit sequence. Indeed, this means that we partition the bits ... more >>>

TR09-017 | 20th February 2009
Joshua Brody

The Maximum Communication Complexity of Multi-party Pointer Jumping.

We study the one-way number-on-the-forhead (NOF) communication complexity of the k-layer pointer jumping problem. Strong lower bounds for this problem would have important implications in circuit complexity. All of our results apply to myopic protocols (where players see only one layer ahead, but can still see arbitrarily far behind them.) ... more >>>

TR09-016 | 21st February 2009
Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio, Emanuele Viola

Bounded Independence Fools Halfspaces

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps)/\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = \Omega(1/(\eps^2 \cdot \log(1/\eps))). Using standard constructions of k-wise ... more >>>

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

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

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

TR09-014 | 7th February 2009
Soren Riis

On the asymptotic Nullstellensatz and Polynomial Calculus proof complexity

We show that the asymptotic complexity of uniformly generated (expressible in First-Order (FO) logic) propositional tautologies for the Nullstellensatz proof system (NS) as well as for Polynomial Calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by ... 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 >>>

TR09-012 | 4th February 2009
Noga Alon, Shai Gutner

Balanced Hashing, Color Coding and Approximate Counting

Color Coding is an algorithmic technique for deciding efficiently if a given input graph contains a path of a given length (or another small subgraph of constant tree-width). Applications of the method in computational biology motivate the study of similar algorithms for counting the number of copies of a given ... more >>>

TR09-011 | 31st January 2009
Mark Braverman

Poly-logarithmic independence fools AC0 circuits

We prove that poly-sized AC0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has ... 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 >>>

TR09-009 | 18th December 2008
Vijayaraghavan T.C.

Checking Equality of Matroid Linear Representations and the Cycle Matching Problem

Revisions: 2

Given linear representations M_1 and M_2 of matroids over a field F, we consider the problem (denoted by ECLR), of checking if M_1 and M_2 represent the same matroid. We show that when F=Z_2, ECLR{Z_2} is complete for $\parityL$. Let M_1,M_2\in Q ^{m\times n} be two matroid linear representations given ... more >>>

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

Min-Rank Conjecture for Log-Depth Circuits

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

TR09-007 | 9th January 2009
Eli Ben-Sasson, Michael Viderman, Michael Viderman

Tensor Products of Weakly Smooth Codes are Robust

We continue the study of {\em robust} tensor codes and expand the class of base codes that can be used as a starting point for the construction of locally testable codes via robust two-wise tensor products. In particular, we show that all unique-neighbor expander codes and all locally correctable codes, ... more >>>

TR09-006 | 19th January 2009
David Xiao

On basing ZK != BPP on the hardness of PAC learning

Learning is a central task in computer science, and there are various formalisms for capturing the notion. One important model studied in computational learning theory is the PAC model of Valiant (CACM 1984). On the other hand, in cryptography the notion of ``learning nothing'' is often modelled by the simulation ... more >>>

TR09-005 | 7th December 2008
Emanuele Viola

Bit-Probe Lower Bounds for Succinct Data Structures

We prove lower bounds on the redundancy necessary to represent a set $S$ of objects using a number of bits close to the information-theoretic minimum $\log_2 |S|$, while answering various queries by probing few bits. Our main results are: \begin{itemize} \item To represent $n$ ternary values $t \in \zot^n$ in ... more >>>

TR09-004 | 15th January 2009
Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

Revisions: 2

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction. \begin{enumerate} \item We show that every Kakeya set in $\F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight ... more >>>

TR09-003 | 6th January 2009
Alex Hertel, Alasdair Urquhart

Comments on ECCC Report TR06-133: The Resolution Width Problem is EXPTIME-Complete

We discovered a serious error in one of our previous submissions to ECCC and wish to make sure that this mistake is publicly known. The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (Exists,k)-pebble game, ... more >>>

TR09-002 | 23rd November 2008
Eli Ben-Sasson, Jakob Nordström

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering ... more >>>

TR09-001 | 26th November 2008
Venkatesan Guruswami

Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes

Algebraic codes that achieve list decoding capacity were recently constructed by a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes ... more >>>

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