Results 1  10
of
13
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
"... In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*ast ..."
Abstract

Cited by 59 (8 self)
 Add to MetaCart
In this paper, we present improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small numberof colors. H*astad's celebrated result [13] shows that the maximumclique size in a graph with n vertices is inapproximable inpolynomial time within a factor n1ffl for arbitrarily smallconstant ffl> 0 unless NP=ZPP. In this paper, we aimat getting the best subconstant value of ffl in H*astad's result. We prove that clique size is inapproximable within a factor n2(log n)1fl (corresponding to ffl = 1(log n)fl) for some constant fl> 0 unless NP ` ZPTIME(2(log n) O(1)). This improves the previous best inapproximability factor of
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
Abstract

Cited by 35 (3 self)
 Add to MetaCart
We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following: 1. 3SAT cannot be efficiently approximated to within a factor of 7 8 + o(1), unless P = N P. This holds even under almostlinear reductions. Previously, the best known N Phardness
Inapproximability of combinatorial optimization problems
 Electronic Colloquium on Computational Complexity
, 2004
"... ..."
Query efficient PCPs with perfect completeness
 In 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR9987077.
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
Constraint Satisfaction over a NonBoolean Domain: Approximation Algorithms and UniqueGames Hardness
 In Proceedings of APPROX 2008
, 2008
"... We study the approximability of the Max kCSP problem over nonboolean domains, more specifically over {0, 1,..., q − 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [18] to obtain a UGC hardness result when q is a prime. More precisely, assuming the Unique Games Conjec ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
We study the approximability of the Max kCSP problem over nonboolean domains, more specifically over {0, 1,..., q − 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [18] to obtain a UGC hardness result when q is a prime. More precisely, assuming the Unique Games Conjecture, we show that it is NPhard to approximate the problem to a ratio greater than q2k/qk. Independent of this work, Austrin and Mossel [2] obtain a more general UGC hardness result using entirely different techniques. We also obtain an approximation algorithm that achieves a ratio of C(q) · k/qk for some constant C(q) depending only on q. Except for constant factors depending on q, the algorithm and the UGC hardness result have the same dependence on the arity k. It has been pointed out to us [14] that a similar approximation ratio can be obtained by reducing the nonboolean case to a boolean CSP, and appealing to the CMM algorithm [3]. As a subroutine, we design a constant factor(depending on q) approximation algorithm for the problem of maximizing a semidefinite quadratic form, where the variables are constrained to take values on the corners of the qdimensional simplex. This result generalizes an algorithm of Nesterov [15] for maximizing semidefinite quadratic forms where the variables take {−1, 1} values.
On an Extremal Hypergraph Problem of Brown, Erdős and Sós
 COMBINATORICA
, 2005
"... Let f r (n, v, e) denote the maximum number of edges in an runiform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemeredi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 19 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Let f r (n, v, e) denote the maximum number of edges in an runiform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemeredi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2 k < r, we have ).
Breaking the ɛSoundness Bound of the Linearity Test over GF(2)
, 2008
"... For Boolean functions that are ɛfar from the set of linear functions, we study the lower bound on the rejection probability (denoted by rej(ɛ)) of the linearity test suggested by Blum, Luby and Rubinfeld. This problem is arguably the most fundamental and extensively studied problem in property test ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
For Boolean functions that are ɛfar from the set of linear functions, we study the lower bound on the rejection probability (denoted by rej(ɛ)) of the linearity test suggested by Blum, Luby and Rubinfeld. This problem is arguably the most fundamental and extensively studied problem in property testing of Boolean functions. The previously best bounds for rej(ɛ) were obtained by Bellare, Coppersmith, H˚astad, Kiwi and Sudan. They used Fourier analysis to show that rej(ɛ) ≥ ɛ for every 0 ≤ ɛ ≤ 1/2. They also conjectured that this bound might not be tight for ɛ’s which are close to 1/2. In this paper we show that this indeed is the case. Specifically, we improve the lower bound of rej(ɛ) ≥ ɛ by an additive constant that depends only on ɛ: rej(ɛ) ≥ ɛ + min{1376ɛ 3 (1 − 2ɛ) 12, 1 4 ɛ(1 − 2ɛ)4}, for every 0 ≤ ɛ ≤ 1/2. Our analysis is based on a relationship between rej(ɛ) and the weight distribution of a coset code of the Hadamard code. We use both Fourier analysis and coding theory tools to estimate this weight distribution.
NonAbelian Homomorphism Testing, and DistributionsClose to their SelfConvolutions
"... Abstract. In this paper, we study two questions related to the problem of testingwhether a function is close to a homomorphism. For two finite groups ..."
Abstract
 Add to MetaCart
Abstract. In this paper, we study two questions related to the problem of testingwhether a function is close to a homomorphism. For two finite groups