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95
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 244 (35 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 125 (10 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.
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 approxima ..."
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Cited by 93 (4 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 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
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 = ˜ ..."
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Cited by 68 (1 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
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 65 (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
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 60 (12 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 &quot;) for arbitrary constants &quot;> 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 &quot;. In additionto being simpler, this proof also works for superconstant values of k up to (log N)1/c where
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
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 56 (9 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
Lowdegree tests at large distances
 In Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... Abstract We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and thenumber of queries. In particular, ..."
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Cited by 50 (2 self)
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Abstract We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and thenumber of queries. In particular, we show that functions with small Gowers uniformity norms behave &quot;randomly &quot; with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem forthe third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficientlythe distance from the secondorder ReedMuller code on inputs lying far beyond its listdecoding radius.