Date of this Version
45th Annual IEEE Symposium on Foundations of Computer Science
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 . This implies that if the Unique Games Conjecture of Khot  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 . A stronger version of this conjecture called Plurality Is Stablest is still open, although  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 . 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 , 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 . 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.
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
Khot, S., Kindler, G., Mossel, E., & O'Donnell, R. (2004). Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?. 45th Annual IEEE Symposium on Foundations of Computer Science, 146-154. http://dx.doi.org/10.1109/FOCS.2004.49
Date Posted: 27 November 2017