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70
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 173 (24 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
Relations Between Average Case Complexity and Approximation Complexity (Extended Abstract)
 In Proceedings of the 34th Annual ACM Symposium on Theory of Computing
, 2002
"... We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardne ..."
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Cited by 89 (9 self)
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We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense ksubgraph, max bipartite clique and the 2catalog segmentation problem. No NPhardness of approximation results are currently known for these problems.
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
NonApproximability Results for Optimization Problems on Bounded Degree Instances
, 2001
"... We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to with ..."
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Cited by 78 (5 self)
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We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor 7=8+O(1= B), unless RP = NP . Hastad [18] proved that the problem is approximable to within a factor 7=8+1=64B in polynomial time, and that is hard to approximate to within a factor 7=8 + 1=(log B) 3 . Our result uses a new randomized reduction from general instances of Max 3SAT to boundedoccurrences instances. The randomized reduction applies to other Max SNP problems as well.
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 59 (8 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
A new multilayered PCP and the hardness of hypergraph vertex cover
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a ..."
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Cited by 53 (10 self)
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Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a factor of ( k 1 ") for arbitrary constants "> 0 and k> = 3. The resultis nearly tight as this problem can be easily approximated within factor k. Our constructionmakes use of the biased LongCode and is analyzed using combinatorial properties of swise tintersecting families of subsets.We also give a different proof that shows an inapproximability factor of b k 2 c ". In additionto being simpler, this proof also works for superconstant values of k up to (log N)1/c where
Gowers uniformity, influence of variables, and PCPs
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We pro ..."
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Cited by 50 (2 self)
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Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimensiond Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that 〈{fS} 〉 U d  ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph longcode test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 1
Hardness of the undirected edgedisjoint paths problem
 Proc. of STOC
, 2005
"... In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n 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 through any edge in the graph. When c ..."
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Cited by 50 (8 self)
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In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n 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 through any edge in the graph. When c = 1, the problem is simply referred to as the EdgeDisjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. We obtain an improved hardness result for EDP, and also show the first polylogarithmic integrality gaps and hardness of approximation results for EDPwC. Specifically, we prove that EDP is (log 1 2 −ε n)hard to approximate for any constant ε> 0, unless NP ⊆ ZP T IME(n polylog n). We also show that for any congestion c = o(log log n / log log log n), there is no (log 1−ε c+1 n)approximation algorithm for EDPwC, unless NP ⊆ ZP T IME(npolylog n). For larger congestion, where c ≤ η log log n / log log log n for some constant η, we obtain superconstant inapproximability ratios. All of our hardness results can be converted into integrality gaps for the multicommodity flow relaxation. We also present a separate elementary direct proof of this integrality gap result. Finally, we note that similar results can be obtained for the AllorNothing Flow (ANF) problem, a relaxation of EDP, in which the flow unit routed between the sourcesink pairs does not have follow a single path, so the resulting flow is not necessarily integral. Using standard transformations, our results also extend to the nodedisjoint versions of these problems as well as to the directed setting. 1
Simple PCPs with Polylog Rate and Query Complexity
, 2005
"... We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constr ..."
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Cited by 47 (15 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constructions of short PCPs (from [5] to [9]) relied extensively on properties of low degree multivariate polynomials. In contrast, our constructions rely on new problems and techniques revolving around the properties of codes based on high degree polynomials in one variable (also known as ReedSolomon codes). We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a ReedSolomon codeword, i.e., a univariate polynomial of specified degree. This reduction is simpler than the corresponding steps in previous reductions, and gives a new alternative to using the popular “sumcheck protocol”. We then give a new PCP for the special task of proving that a function is close to being a ReedSolomon codeword. This step of the construction is by a selfcontained recursion, and the only ingredient needed in the analysis is the bivariate lowdegree test of Polischuk and Spielman [27]. Note that our constructions yield LTCs first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite (and less natural) direction of getting LTCs from PCPs.
Linear degree extractors and the inapproximability of max clique and chromatic number
 THEORY OF COMPUTING
, 2007
"... ... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜P. The ..."
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Cited by 46 (0 self)
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... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜P. The key to these results is a new construction of dispersers, which are related to randomness extractors. A randomness extractor is an algorithm which extracts randomness from a lowquality random source, using some additional truly random bits. We construct new extractors which require only log2 n + O(1) additional random bits for sources with constant entropy rate, and have constant error. Our dispersers use an arbitrarily small constant