Statistics Papers

Document Type

Conference Paper

Date of this Version

10-2004

Publication Source

45th Annual IEEE Symposium on Foundations of Computer Science

Start Page

146

Last Page

154

DOI

10.1109/FOCS.2004.49

Abstract

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of αGW + ∈, for all ∈ > 0; here αGW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT 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 [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it.

Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q).

For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [25], namely 1 − 1/q + 2(ln q)/q2 .

For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−∈)-satisfiable and those which are not even, roughly, (q−∈/2)-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for instances in which all equations are of the form xi − xj = c. At a more qualitative level, this result also implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture.

Keywords

approximation theory, communicating sequential processes, computability, computational complexity, game theory, graph theory, optimisation, CSP, Goemans-Williamson algorithm, MAX-2CSP problem, MAX-2SAT problem, MAX-CUT, NP-hard, approximation algorithm, games conjecture, majority is stablest conjecture, nonBoolean domains, optimal inapproximability, additive noise, approximation algorithms, bipartite graph, Boolean functions, computer science, labeling, mathematics, stability, statistics

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Date Posted: 27 November 2017