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REPORTS > AUTHORS > VENKATESAN GURUSWAMI:
All reports by Author Venkatesan Guruswami:

TR23-126 | 25th August 2023
Omar Alrabiah, Venkatesan Guruswami, Ray Li

AG codes have no list-decoding friends: Approaching the generalized Singleton bound requires exponential alphabets

A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for ... more >>>


TR23-125 | 25th August 2023
Omar Alrabiah, Venkatesan Guruswami, Ray Li

Randomly punctured Reed-Solomon codes achieve list-decoding capacity over linear-sized fields

Reed-Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a ... more >>>


TR21-145 | 19th October 2021
Omar Alrabiah, Venkatesan Guruswami

Revisiting a Lower Bound on the Redundancy of Linear Batch Codes

A recent work of Li and Wootters (2021) shows a redundancy lower bound of $\Omega(\sqrt{Nk})$ for systematic linear $k$-batch codes of block length $N$ by looking at the $O(k)$ tensor power of the dual code. In this note, we present an alternate proof of their result via a linear independence ... more >>>


TR21-139 | 24th September 2021
Venkatesan Guruswami, Jonathan Mosheiff

Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

Revisions: 2

We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb{F}_q$ that is polynomially larger than the ... more >>>


TR21-119 | 13th August 2021
Omar Alrabiah, Venkatesan Guruswami

Visible Rank and Codes with Locality

Revisions: 2

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call "visible rank." The locality constraints of a linear code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we ... more >>>


TR21-112 | 30th July 2021
Vikraman Arvind, Venkatesan Guruswami

CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of ... more >>>


TR21-079 | 9th June 2021
Venkatesan Guruswami, Xiaoyu He, Ray Li

The zero-rate threshold for adversarial bit-deletions is less than 1/2

We prove that there exists an absolute constant $\delta>0$ such any binary code $C\subset\{0,1\}^N$ tolerating $(1/2-\delta)N$ adversarial deletions must satisfy $|C|\le 2^{\poly\log N}$ and thus have rate asymptotically approaching $0$. This is the first constant fraction improvement over the trivial bound that codes tolerating $N/2$ adversarial deletions must have rate ... more >>>


TR21-026 | 23rd February 2021
Joshua Brakensiek, Venkatesan Guruswami, Sai Sandeep

Conditional Dichotomy of Boolean Ordered Promise CSPs

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their ... more >>>


TR21-025 | 15th February 2021
Sivakanth Gopi, Venkatesan Guruswami

Improved Maximally Recoverable LRCs using Skew Polynomials

An $(n,r,h,a,q)$-Local Reconstruction Code is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity ... more >>>


TR20-172 | 13th November 2020
Venkatesan Guruswami, Chaoping Xing

Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

We construct two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-\epsilon$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\epsilon$. The alphabet size depends only on $\epsilon$ and is nearly-optimal.

The ... more >>>


TR20-167 | 9th November 2020
Venkatesan Guruswami, Sai Sandeep

Approximate Hypergraph Vertex Cover and generalized Tuza's conjecture

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this ... more >>>


TR20-151 | 8th October 2020
Venkatesan Guruswami, Vinayak Kumar

Pseudobinomiality of the Sticky Random Walk

Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number $M$ of marked vertices visited in a long $n$-step random walk strongly concentrates around the ... more >>>


TR20-004 | 17th January 2020
Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Zivny

The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

Revisions: 1

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not ... more >>>


TR19-154 | 6th November 2019
Venkatesan Guruswami, Andrii Riazanov, Min Ye

Ar?kan meets Shannon: Polar codes with near-optimal convergence to channel capacity

Revisions: 3

Let $W$ be a binary-input memoryless symmetric (BMS) channel with Shannon capacity $I(W)$ and fix any $\alpha > 0$. We construct, for any sufficiently small $\delta > 0$, binary linear codes of block length $O(1/\delta^{2+\alpha})$ and rate $I(W)-\delta$ that enable reliable communication on $W$ with quasi-linear time encoding and decoding. ... more >>>


TR19-153 | 6th November 2019
Venkatesan Guruswami, Bernhard Haeupler, Amirbehshad Shahrasbi

Optimally Resilient Codes for List-Decoding from Insertions and Deletions

Revisions: 1

We give a complete answer to the following basic question: ``What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?''

This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best known results ... more >>>


TR19-116 | 9th September 2019
Venkatesan Guruswami, Sai Sandeep

$d$-to-$1$ Hardness of Coloring $4$-colorable Graphs with $O(1)$ colors

Revisions: 1

The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ ... more >>>


TR19-094 | 16th July 2019
Venkatesan Guruswami, Sai Sandeep

Rainbow coloring hardness via low sensitivity polymorphisms

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>


TR19-092 | 9th July 2019
Venkatesan Guruswami, Jakub Opršal, Sai Sandeep

Revisiting Alphabet Reduction in Dinur's PCP

Dinur's celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the ... more >>>


TR19-054 | 9th April 2019
Joshua Brakensiek, Venkatesan Guruswami

Bridging between 0/1 and Linear Programming via Random Walks

Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial ... more >>>


TR19-013 | 31st January 2019
Joshua Brakensiek, Sivakanth Gopi, Venkatesan Guruswami

CSPs with Global Modular Constraints: Algorithms and Hardness via Polynomial Representations

We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo $M$, for various choices of the modulus $M$. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2SAT, HornSAT, ... more >>>


TR19-005 | 16th January 2019
Omar Alrabiah, Venkatesan Guruswami

An Exponential Lower Bound on the Sub-Packetization of MSR Codes

Revisions: 1

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a ... more >>>


TR18-097 | 15th May 2018
Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

Approximating Operator Norms via Generalized Krivine Rounding

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A \in {\mathbb R}^{m \times n}$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^\ast$ operator norm of $A$, where $\ell_{r^*}$ is the dual norm ... more >>>


TR18-096 | 13th May 2018
Venkatesan Guruswami, Andrii Riazanov

Beating Fredman-Komlós for perfect $k$-hashing

We say a subset $C \subseteq \{1,2,\dots,k\}^n$ is a $k$-hash code (also called $k$-separated) if for every subset of $k$ codewords from $C$, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as $(\log_2 |C|)/n$, of a $k$-hash code is ... more >>>


TR18-059 | 6th April 2018
Joshua Brakensiek, Venkatesan Guruswami

Combining LPs and Ring Equations via Structured Polymorphisms

Revisions: 1

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well known problems can be cast as promise CSPs including approximate graph and hypergraph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, ... more >>>


TR18-037 | 21st February 2018
Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

Inapproximability of Matrix $p \rightarrow q$ Norms

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been ... more >>>


TR18-027 | 8th February 2018
Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, Madhu Sudan

General Strong Polarization

Revisions: 1

Ar\i kan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the $\textit{polarization}$ of an associated $[0,1]$-bounded martingale, ... 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 >>>


TR17-183 | 28th November 2017
Sivakanth Gopi, Venkatesan Guruswami, Sergey Yekhanin

On Maximally Recoverable Local Reconstruction Codes

In recent years the explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. An $(n,r,h,a,q)$-LRC is a $q$-ary code, where encoding is as a ... more >>>


TR17-147 | 3rd October 2017
Venkatesan Guruswami, Rishi Saket

Hardness of Rainbow Coloring Hypergraphs

A hypergraph is $k$-rainbow colorable if there exists a vertex coloring using $k$ colors such that each hyperedge has all the $k$ colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be ... more >>>


TR17-141 | 19th September 2017
Joshua Brakensiek, Venkatesan Guruswami

A Family of Dictatorship Tests with Perfect Completeness for 2-to-2 Label Cover

We give a family of dictatorship tests with perfect completeness and low-soundness for 2-to-2 constraints. The associated 2-to-2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. ... more >>>


TR17-080 | 1st May 2017
Joshua Brakensiek, Venkatesan Guruswami

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture (UGC) has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect ... more >>>


TR17-064 | 20th April 2017
Venkatesan Guruswami, Chaoping Xing, chen yuan

Subspace Designs based on Algebraic Function Fields

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The ... more >>>


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

Multiplicative Approximations for Polynomial Optimization Over the Unit Sphere

Revisions: 1

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


TR16-183 | 16th November 2016
Joshua Brakensiek, Venkatesan Guruswami

Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy

Revisions: 2

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or NP-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, ... more >>>


TR16-033 | 10th March 2016
Venkatesan Guruswami, Jaikumar Radhakrishnan

Tight bounds for communication assisted agreement distillation

Suppose Alice holds a uniformly random string $X \in \{0,1\}^N$ and Bob holds a noisy version $Y$ of $X$ where each bit of $X$ is flipped independently with probability $\epsilon \in [0,1/2]$. Alice and Bob would like to extract a common random string of min-entropy at least $k$. In this ... more >>>


TR16-029 | 7th March 2016
Joshua Brakensiek, Venkatesan Guruswami

New hardness results for graph and hypergraph colorings

Finding a proper coloring of a $t$-colorable graph $G$ with $t$ colors is a classic NP-hard problem when $t\ge 3$. In this work, we investigate the approximate coloring problem in which the objective is to find a proper $c$-coloring of $G$ where $c \ge t$. We show that for all ... more >>>


TR15-155 | 22nd September 2015
Venkatesan Guruswami, Euiwoong Lee

Nearly Optimal NP-Hardness of Unique Coverage

The {\em Unique Coverage} problem, given a universe $V$ of elements and a collection $E$ of subsets of $V$, asks to find $S \subseteq V$ to maximize the number of $e \in E$ that intersects $S$ in {\em exactly one} element. When each $e \in E$ has cardinality at most ... more >>>


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

An improved bound on the fraction of correctable deletions

Revisions: 1

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


TR15-116 | 21st July 2015
Joshua Brakensiek, Venkatesan Guruswami, Samuel Zbarsky

Efficient Low-Redundancy Codes for Correcting Multiple Deletions

Revisions: 1

We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ ... more >>>


TR15-105 | 21st June 2015
Venkatesan Guruswami, Euiwoong Lee

Towards a Characterization of Approximation Resistance for Symmetric CSPs

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to $1$ with some probability $\alpha$ achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), ... more >>>


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

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

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


TR14-162 | 28th November 2014
Michael Forbes, Venkatesan Guruswami

Dimension Expanders via Rank Condensers

An emerging theory of "linear-algebraic pseudorandomness" aims to understand the linear-algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension ... more >>>


TR14-153 | 14th November 2014
Clement Canonne, Venkatesan Guruswami, Raghu Meka, Madhu Sudan

Communication with Imperfectly Shared Randomness

Revisions: 3

The communication complexity of many fundamental problems reduces greatly
when the communicating parties share randomness that is independent of the
inputs to the communication task. Natural communication processes (say between
humans) however often involve large amounts of shared correlations among the
communicating players, but rarely allow for perfect sharing of ... 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 >>>


TR14-043 | 2nd April 2014
Venkatesan Guruswami, Euiwoong Lee

Strong Inapproximability Results on Balanced Rainbow-Colorable Hypergraphs

Consider a $K$-uniform hypergraph $H = (V,E)$. A coloring $c : V \rightarrow \{1, 2, \dots, k \}$ with $k$ colors is rainbow if every hyperedge $e$ contains at least one vertex from each color, and is called perfectly balanced when each color appears the same number of times. A ... more >>>


TR14-006 | 16th January 2014
Venkatesan Guruswami, Euiwoong Lee

Inapproximability of Feedback Vertex Set for Bounded Length Cycles

Revisions: 1

The Feedback Vertex Set problem (FVS), where the goal is to find a small subset of vertices that intersects every cycle in an input directed graph, is among the fundamental problems whose approximability is not well-understood. One can efficiently find an $\widetilde{O}(\log n)$ factor approximation, and while a constant-factor approximation ... more >>>


TR13-175 | 6th December 2013
Venkatesan Guruswami, Chaoping Xing

Hitting Sets for Low-Degree Polynomials with Optimal Density

Revisions: 1

We give a length-efficient puncturing of Reed-Muller codes which preserves its distance properties. Formally, for the Reed-Muller code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ with $q \ge \Omega(d/\delta)$, we present an explicit (multi)-set $S \subseteq {\mathbb F}_q^n$ of size $N=\mathrm{poly}(n^d/\delta)$ such that every nonzero polynomial vanishes on at most ... more >>>


TR13-170 | 2nd December 2013
Venkatesan Guruswami, Carol Wang

Explicit rank-metric codes list-decodable with optimal redundancy

We construct an explicit family of linear rank-metric codes over any field ${\mathbb F}_h$ that enables efficient list decoding up to a fraction $\rho$ of errors in the rank metric with a rate of $1-\rho-\epsilon$, for any desired $\rho \in (0,1)$ and $\epsilon > 0$. Previously, a Monte Carlo construction ... more >>>


TR13-167 | 28th November 2013
Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, Girish Varma

Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on ... more >>>


TR13-159 | 20th November 2013
Per Austrin, Venkatesan Guruswami, Johan Håstad

$(2+\epsilon)$-SAT is NP-hard

Revisions: 2

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find ... more >>>


TR13-125 | 11th September 2013
Venkatesan Guruswami, Euiwoong Lee

Complexity of approximating CSP with Balance / Hard Constraints

We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize ... more >>>


TR13-122 | 5th September 2013
Irit Dinur, Venkatesan Guruswami

PCPs via low-degree long code and hardness for constrained hypergraph coloring

Revisions: 1

We develop new techniques to incorporate the recently proposed ``short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical ``Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>>


TR13-121 | 4th September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Non-Malleable Coding Against Bit-wise and Split-State Tampering

Revisions: 1

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>


TR13-118 | 2nd September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Capacity of Non-Malleable Codes

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly ... more >>>


TR13-071 | 8th May 2013
Venkatesan Guruswami, Sushant Sachdeva, Rishi Saket

Inapproximability of Minimum Vertex Cover on $k$-uniform $k$-partite Hypergraphs

We study the problem of computing the minimum vertex cover on $k$-uniform $k$-partite hypergraphs when the $k$-partition is given. On bipartite graphs ($k=2$), the minimum vertex cover can be computed in polynomial time. For $k \ge 3$, this problem is known to be NP-hard. For general $k$, the problem was ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-050 | 1st April 2013
Venkatesan Guruswami, Patrick Xia

Polar Codes: Speed of polarization and polynomial gap to capacity

Revisions: 1

We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $\epsilon > 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a *polynomial* in $1/\epsilon$. Polar coding gives the *first known explicit construction* with rigorous ... more >>>


TR13-046 | 27th March 2013
Venkatesan Guruswami, Chaoping Xing

Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets

We construct a new list-decodable family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. The function fields underlying these codes are constructed using class field theory, specifically Drinfeld modules of rank $1$, and designed to have an automorphism of large order that is used to ``fold" the AG code. ... more >>>


TR13-002 | 31st December 2012
Venkatesan Guruswami, Krzysztof Onak

Superlinear lower bounds for multipass graph processing

Revisions: 3

We prove $n^{1+\Omega(1/p)}/p^{O(1)}$ lower bounds for the space complexity of $p$-pass streaming algorithms solving the following problems on $n$-vertex graphs:

* testing if an undirected graph has a perfect matching (this implies lower bounds for computing a maximum matching or even just the maximum matching size),

* testing if two ... 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 >>>


TR12-111 | 5th September 2012
Venkatesan Guruswami, Ali Kemal Sinop

Faster SDP hierarchy solvers for local rounding algorithms

Convex relaxations based on different hierarchies of
linear/semi-definite programs have been used recently to devise
approximation algorithms for various optimization problems. The
approximation guarantee of these algorithms improves with the number
of {\em rounds} $r$ in the hierarchy, though the complexity of solving
(or even writing down the solution for) ... more >>>


TR12-082 | 28th June 2012
Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-\epsilon$ with list size $L=O(1/\epsilon^2)$ and rate $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to the polylogarithmic factor in $1/\epsilon$ and constant factors depending on $q$, this matches the lower bound $L=\Omega_q(1/\epsilon^2)$ for the list ... more >>>


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

Approximating Bounded Occurrence Ordering CSPs

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


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 >>>


TR12-036 | 12th April 2012
Venkatesan Guruswami, Chaoping Xing

Folded Codes from Function Field Towers and Improved Optimal Rate List Decoding

We give a new construction of algebraic codes which are efficiently list decodable from a fraction $1-R-\epsilon$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\epsilon$. The worst-case list size output by the algorithm is $O(1/\epsilon)$, matching the existential bound for random ... more >>>


TR12-017 | 1st March 2012
Venkatesan Guruswami, Srivatsan Narayanan

Combinatorial limitations of a strong form of list decoding

Revisions: 1

We prove the following results concerning the combinatorics of list decoding, motivated by the exponential gap between the known upper bound (of $O(1/\gamma)$) and lower bound (of $\Omega_p(\log (1/\gamma))$) for the list-size needed to decode up to radius $p$ with rate $\gamma$ away from capacity, i.e., $1-h(p)-\gamma$ (here $p\in (0,1/2)$ ... 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 >>>


TR11-027 | 28th February 2011
Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, Moses Charikar

Beating the Random Ordering is Hard: Every ordering CSP is approximation resistant

We prove that, assuming the Unique Games Conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSP) where each constraint has constant arity is approximation
resistant. In other words, we show that if $\rho$ is the expected fraction of constraints satisfied by a random ordering, then obtaining ... more >>>


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

Agnostic Learning of Monomials by Halfspaces is Hard

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


TR10-177 | 16th November 2010
Venkatesan Guruswami, Prasad Raghavendra, Rishi Saket, Yi Wu

Bypassing UGC from some optimal geometric inapproximability results

Revisions: 1

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>


TR10-111 | 14th July 2010
Venkatesan Guruswami, Ali Kemal Sinop

The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number

We prove almost tight hardness results for finding independent sets in bounded degree graphs and hypergraphs that admit a good
coloring. Our specific results include the following (where $\Delta$, assumed to be a constant, is a bound on the degree, and
$n$ is the number of vertices):

... more >>>

TR10-077 | 26th April 2010
Venkatesan Guruswami, Adam Smith

Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate

In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently ... more >>>


TR10-063 | 12th April 2010
Venkatesan Guruswami, Yuan Zhou

Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set}

Revisions: 1

We study the approximability of two natural Boolean constraint satisfaction problems: Horn satisfiability and exact hitting set. Under the Unique Games conjecture, we prove the following optimal inapproximability and approximability results for finding an assignment satisfying as many constraints as possible given a {\em
near-satisfiable} instance.

\begin{enumerate}
\item ... more >>>


TR10-003 | 6th January 2010
Venkatesan Guruswami, Johan Håstad, Swastik Kopparty

On the List-Decodability of Random Linear Codes

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >
0$, we prove that with high probability a random subspace $C$ of
$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the
property that every Hamming ball of radius $pn$ has at most
$O(1/\varepsilon)$ codewords.

This ... 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 >>>


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 >>>


TR09-020 | 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 >>>


TR09-001 | 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 >>>


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 >>>


TR08-054 | 13th May 2008
Venkatesan Guruswami, Atri Rudra

Concatenated codes can achieve list-decoding capacity

We prove that binary linear concatenated codes with an outer algebraic code (specifically, a folded Reed-Solomon code) and independently and randomly chosen linear inner codes achieve the list-decoding capacity with high probability. In particular, for any $0 < \rho < 1/2$ and $\epsilon > 0$, there exist concatenated codes of ... more >>>


TR08-036 | 14th March 2008
Venkatesan Guruswami, Atri Rudra

Soft decoding, dual BCH codes, and better list-decodable eps-biased codes

We construct binary linear codes that are efficiently list-decodable
up to a fraction $(1/2-\eps)$ of errors. The codes encode $k$ bits
into $n = {\rm poly}(k/\eps)$ bits and are constructible and
list-decodable in time polynomial in $k$ and $1/\eps$ (in
particular, in our results $\eps$ need ... more >>>


TR08-008 | 8th February 2008
Venkatesan Guruswami, Prasad Raghavendra

Constraint Satisfaction over a Non-Boolean Domain: Approximation algorithms and Unique-Games hardness

Revisions: 1

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>


TR07-113 | 15th November 2007
Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, Lisa Zhang

Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs

In the undirected Edge-Disjoint Paths problem with Congestion
(EDPwC), we are given an undirected graph with $V$ nodes, a set of
terminal pairs and an integer $c$. The objective is to route as many
terminal pairs as possible, subject to the constraint that at most
$c$ demands can be routed ... more >>>


TR07-109 | 7th October 2007
Venkatesan Guruswami, Atri Rudra

Better Binary List-Decodable Codes via Multilevel Concatenation

We give a polynomial time construction of binary codes with the best
currently known trade-off between rate and error-correction
radius. Specifically, we obtain linear codes over fixed alphabets
that can be list decoded in polynomial time up to the so called
Blokh-Zyablov bound. Our work ... more >>>


TR07-089 | 13th September 2007
Parikshit Gopalan, Venkatesan Guruswami

Deterministic Hardness Amplification via Local GMD Decoding

We study the average-case hardness of the class NP against
deterministic polynomial time algorithms. We prove that there exists
some constant $\mu > 0$ such that if there is some language in NP
for which no deterministic polynomial time algorithm can decide L
correctly on a $1- (log n)^{-\mu}$ fraction ... more >>>


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

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

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


TR06-141 | 22nd November 2006
Venkatesan Guruswami, Kunal Talwar

Hardness of Low Congestion Routing in Directed Graphs

We prove a strong inapproximability result for routing on directed
graphs with low congestion. Given as input a directed graph on $N$
vertices and a set of source-destination pairs that can be connected
via edge-disjoint paths, we prove that it is hard, assuming NP
doesn't have $n^{O(\log\log n)}$ time randomized ... more >>>


TR06-134 | 18th October 2006
Venkatesan Guruswami, Chris Umans, Salil Vadhan

Extractors and condensers from univariate polynomials

Revisions: 1

We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>


TR06-123 | 15th September 2006
Venkatesan Guruswami, Venkatesan Guruswami

Iterative Decoding of Low-Density Parity Check Codes (A Survey)

Much progress has been made on decoding algorithms for
error-correcting codes in the last decade. In this article, we give an
introduction to some fundamental results on iterative, message-passing
algorithms for low-density parity check codes. For certain
important stochastic channels, this line of work has enabled getting
very close to ... more >>>


TR06-123 | 15th September 2006
Venkatesan Guruswami, Venkatesan Guruswami

Iterative Decoding of Low-Density Parity Check Codes (A Survey)

Much progress has been made on decoding algorithms for
error-correcting codes in the last decade. In this article, we give an
introduction to some fundamental results on iterative, message-passing
algorithms for low-density parity check codes. For certain
important stochastic channels, this line of work has enabled getting
very close to ... more >>>


TR06-061 | 5th May 2006
Venkatesan Guruswami, Prasad Raghavendra

Hardness of Learning Halfspaces with Noise

Learning an unknown halfspace (also called a perceptron) from
labeled examples is one of the classic problems in machine learning.
In the noise-free case, when a halfspace consistent with all the
training examples exists, the problem can be solved in polynomial
time using linear programming. ... more >>>


TR05-133 | 17th November 2005
Venkatesan Guruswami, Atri Rudra

Explicit Capacity-Achieving List-Decodable Codes

Revisions: 1

For every $0 < R < 1$ and $\eps > 0$, we present an explicit
construction of error-correcting codes of rate $R$ that can be list
decoded in polynomial time up to a fraction $(1-R-\eps)$ of errors.
These codes achieve the ``capacity'' for decoding from {\em ... more >>>


TR05-132 | 8th November 2005
Venkatesan Guruswami

Algebraic-geometric generalizations of the Parvaresh-Vardy codes

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


TR05-057 | 19th May 2005
Venkatesan Guruswami, Valentine Kabanets

Hardness amplification via space-efficient direct products

We prove a version of the derandomized Direct Product Lemma for
deterministic space-bounded algorithms. Suppose a Boolean function
$g:\{0,1\}^n\to\{0,1\}$ cannot be computed on more than $1-\delta$
fraction of inputs by any deterministic time $T$ and space $S$
algorithm, where $\delta\leq 1/t$ for some $t$. Then, for $t$-step
walks $w=(v_1,\dots, v_t)$ ... more >>>


TR05-019 | 9th February 2005
Venkatesan Guruswami, Atri Rudra

Tolerant Locally Testable Codes

An error-correcting code is said to be {\em locally testable} if it has an
efficient spot-checking procedure that can distinguish codewords
from strings that are far from every codeword, looking at very few
locations of the input in doing so. Locally testable codes (LTCs) have
generated ... more >>>


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

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

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


TR03-080 | 12th November 2003
Venkatesan Guruswami

Better Extractors for Better Codes?

We present an explicit construction of codes that can be list decoded
from a fraction $(1-\eps)$ of errors in sub-exponential time and which
have rate $\eps/\log^{O(1)}(1/\eps)$. This comes close to the optimal
rate of $\Omega(\eps)$, and is the first sub-exponential complexity
construction to beat the rate of $O(\eps^2)$ achieved by ... more >>>


TR02-053 | 20th July 2002
Lars Engebretsen, Venkatesan Guruswami

Is Constraint Satisfaction Over Two Variables Always Easy?

By the breakthrough work of Håstad, several constraint satisfaction
problems are now known to have the following approximation resistance
property: satisfying more clauses than what picking a random
assignment would achieve is NP-hard. This is the case for example for
Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable ... more >>>


TR02-027 | 30th April 2002
Irit Dinur, Venkatesan Guruswami, Subhash Khot

Vertex Cover on k-Uniform Hypergraphs is Hard to Approximate within Factor (k-3-\epsilon)

Given a $k$-uniform hypergraph, the E$k$-Vertex-Cover problem is
to find a minimum subset of vertices that ``hits'' every edge. We
show that for every integer $k \geq 5$, E$k$-Vertex-Cover is
NP-hard to approximate within a factor of $(k-3-\epsilon)$, for
an arbitrarily small constant $\epsilon > 0$.

This almost matches the ... more >>>


TR01-002 | 6th December 2000
Venkatesan Guruswami

Constructions of Codes from Number Fields

We define number-theoretic error-correcting codes based on algebraic
number fields, thereby providing a generalization of Chinese Remainder
Codes akin to the generalization of Reed-Solomon codes to
Algebraic-geometric codes. Our construction is very similar to
(and in fact less general than) the one given by (Lenstra 1986), but
the ... more >>>


TR00-073 | 28th August 2000
Venkatesan Guruswami, Sanjeev Khanna

On the Hardness of 4-coloring a 3-colorable Graph

We give a new proof showing that it is NP-hard to color a 3-colorable
graph using just four colors. This result is already known (Khanna,
Linial, Safra 1992), but our proof is novel as it does not rely on
the PCP theorem, while the earlier one does. This ... more >>>


TR00-062 | 25th August 2000
Venkatesan Guruswami, Johan Håstad, Madhu Sudan

Hardness of approximate hypergraph coloring

We introduce the notion of covering complexity of a probabilistic
verifier. The covering complexity of a verifier on a given input is
the minimum number of proofs needed to ``satisfy'' the verifier on
every random string, i.e., on every random string, at least one of the
given proofs must be ... more >>>


TR99-043 | 5th November 1999
Venkatesan Guruswami

The Approximability of Set Splitting Problems and Satisfiability Problems with no Mixed Clauses


We prove hardness results for approximating set splitting problems and
also instances of satisfiability problems which have no ``mixed'' clauses,
i.e all clauses have either all their literals unnegated or all of them
negated. Recent results of Hastad imply tight hardness results for set
splitting when all sets ... more >>>


TR98-043 | 27th July 1998
Venkatesan Guruswami, Madhu Sudan

Improved decoding of Reed-Solomon and algebraic-geometric codes.

We present an improved list decoding algorithm for decoding
Reed-Solomon codes. Given an arbitrary string of length n, the
list decoding problem is that of finding all codewords within a
specified Hamming distance from the input string.

It is well-known that this decoding problem for Reed-Solomon
codes reduces to the ... more >>>


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

A tight characterization of NP with 3 query PCPs


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




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