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17
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 ..."
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Cited by 75 (9 self)
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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 ..."
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Cited by 57 (6 self)
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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
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 ..."
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Cited by 53 (2 self)
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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.
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 probabil ..."
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Cited by 21 (3 self)
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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.
On the communication and streaming complexity of maximum bipartite matching
, 2011
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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 ..."
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Cited by 9 (1 self)
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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.
New hardness results for undirected edge disjoint paths
, 2005
"... In the edgedisjoint paths (EDP) problem, we are given a graph G and a set of sourcesink pairs in G. The goal is connect as many pairs as possible in an edgedisjoint manner. This problem is NPhard and the best known approximation algorithm gives an Õ(min{n2/3, √ m})approximation for both direct ..."
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Cited by 4 (2 self)
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In the edgedisjoint paths (EDP) problem, we are given a graph G and a set of sourcesink pairs in G. The goal is connect as many pairs as possible in an edgedisjoint manner. This problem is NPhard and the best known approximation algorithm gives an Õ(min{n2/3, √ m})approximation for both directed and undirected graphs; here n and m denote the number of vertices and edges in G respectively. For directed graphs, this result is tight as a function of m since it is known that directed EDP is NPhard to approximate to within Ω(m 1/2−ɛ) for any ɛ> 0. However, for undirected graphs, until recently nothing better than APXhardness was known. In a significant improvement, Andrews and Zhang [1] showed that undirected EDP is Ω(log 1/3−ɛ n)hard to approximate unless NP is contained in ZPTIME(n polylog(n)). In this paper, we improve the hardness result of [1] as well as obtain the first polylogarithmic integrality gaps and hardness results for undirected EDP when congestion is allowed. A solution to EDP has congestion c if we allow up to c paths to share an edge. When no congestion is allowed, we establish an Ω(log 1/2−ɛ n)hardness for EDP. With congestion c, we show that the natural multicommodity flow log n relaxation of EDP has an Ω(( (log log n) 2) 1/(c+1) /c) integrality gap. Finally, we show that it is possible to obtain ( a hardness result that is comparable to the integrality gap. In particular, we show that EDP is Ω (log n) (1−ɛ)/ ( 3 2 c+ 1 2)hard to approximate for any constant ɛ> 0, when congestion c is allowed, for any c = o(log log n)/(log log log n) 2, such that c = 2 z − 1 for some integer z. We also obtain superconstant hardness when c is as large as O(log log n)/(log log log n) 2. Similar results can be obtained for the AllorNothing flow problem, a relaxation of EDP in that the unit flow between each routed sourcesink pair does not have to be on a single path. Using standard transformations, these results can also be extended to the nodedisjoint versions of these problems as well as to the directed setting. 1
Nearly Complete Graphs Decomposable into Large Induced Matchings and their Applications
, 2013
"... We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with () N 2 2 − o(N) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N 1−o(1). The second constru ..."
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Cited by 3 (0 self)
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We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with () N 2 2 − o(N) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N 1−o(1). The second construction provides a covering of all edges of the complete graph KN by two graphs, each being the edge disjoint union of at most N 2−δ induced matchings, where δ> 0.076. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of H˚astad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.
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 ..."
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Cited by 2 (2 self)
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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 ).