Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?

• Authors: Subhash Khot;Guy Kindler;Elchanan Mossel;Ryan O’Donnell
• Venue: SIAM Journal on Computing
• Year: 2007

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of $\alpha_{\text{\tiny{GW}}} + \epsilon$ for all $\epsilon 0$; here $\alpha_{\text{\tiny{GW}}} \approx .878567$ denotes the approximation ratio achieved by the algorithm of Goemans and Williamson in [J. Assoc. Comput. Mach., 42 (1995), pp. 1115-1145]. This implies that if the Unique Games Conjecture of Khot in [Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775] 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 [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21-30]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21-30] 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 in [Proceedings of the 9th Annual Conference on Integer Programming and Combinatorial Optimization, Springer-Verlag, Berlin, 2002, pp. 67-82]. 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 [Improved approximation algorithms for MAX k-CUT and MAX BISECTION, in Integer Programming and Combinatorial Optimization, Springer-Verlag, Berlin, pp. 1-13], namely $1 - 1/q + 2({\rm ln}\,q)/q^2$. For MAX-2LIN($q$) we show hardness of distinguishing between instances which are $(1-\epsilon)$-satisfiable and those which are not even, roughly, $(q^{-\epsilon/2})$-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 205-214]. The hardness result holds even for instances in which all equations are of the form $x_i - x_j = c$. At a more qualitative level, this result also implies that $1-\epsilon$ vs. &egr; hardness for MAX-2LIN($q$) is equivalent to the Unique Games Conjecture.