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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 (26 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
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = ..."
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Cited by 37 (12 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].
Proof of a Hypercontractive Estimate via Entropy
, 2001
"... with the uniform (=product) measure. ..."
Noninteractive correlation distillation, inhomogeneous Markov chains, and the reverse BonamiBeckner inequality
 Israel Journal of Mathematics
"... In this paper we study noninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly ra ..."
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Cited by 5 (2 self)
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In this paper we study noninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: • In the case of a kleaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero as k → ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowlydecaying polynomial). • In the case of the kvertex path graph, we show that it is always optimal for all players to use the same 1bit function. • In the general case we show that all players should use monotone functions. We also show, somewhat
Optimal Inapproximability Results for MAXCUT and . . .
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of αGW + ɛ, for all ɛ> 0; here αGW ≈.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games Conjecture o ..."
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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 αGW + ɛ, for all ɛ> 0; here αGW ≈.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games Conjecture of Khot [36] holds then the GoemansWilliamson approximation algorithm is optimal. Our result indicates that the geometric nature of the GoemansWilliamson algorithm might be intrinsic to the MAXCUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [42]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [42] contains a proof of an asymptotic version of it. Our techniques extend to several other twovariable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX2SAT, MAXqCUT, and MAX2LIN(q). For MAX2SAT we show approximation hardness up to a factor of roughly.943. This nearly matches the.940 approximation algorithm of Lewin, Livnat, and Zwick [40]. Furthermore, we show that our.943... factor is actually tight for a slightly restricted version of MAX2SAT. For MAXqCUT we show