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

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 α GW + ∈, for all ∈ > 0; here α GW ≈ .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)/q 2 .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 x i − x j = 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. AbstractIn this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of α GW + , for all > 0; here α GW ≈ .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, M...

Standard simplices and pluralities are not the most noise stable

“…The Original Conjectures. We note again that our results do not contradict the precise conjectures stated in [17,15] as these are stated in the case where the measures of all partition elements are exactly 1/k. However, the authors of [17,15] gave no indication that they believe there is something special about the case of equal measured partitions -rather, they made the conjectures that were needed for the applications presented in their paper.…”

“…Furthermore it was shown in [15] that the standard simplex being optimal for Gaussian noise stability is equivalent to the fact that plurality is stablest. Therefore, the authors of [17,15] conjectured that plurality is stablest and demonstrated a number of applications of this result in hardness of approximation and voting.…”

“…The relevant question in this setting involves lowinfluence partitions. Using a non-linear invariance principle it was shown that the majority function maximizes noise stability [24] as conjectured both in the context of hardness of approximation [17] and in voting [16]. Since plurality is the natural generalization of majority, it was conjectured in [17] that plurality is the most stable low-influence partition into k ≥ 3 parts of {1, .…”

Approximate Graph Isomorphism

“…Since, assuming the Unique Games Conjecture (UGC for short) of Khot [15], it is NP-hard to approximate Max-2Lin-2 beyond a factor of 0.878 [16], the same bound holds under UGC for Max-PGI k and Max-EGI k by the same reduction. Since Max-PGI k and Max-EGI k are easily seen to be instances of generalized 2CSP, they have constant factor approximation algorithms, for a constant factor depending on k. In fact, it turns out that Max-EGI 2 and Max-PGI 2 are tightly classified by Max-2Lin-2 with almost matching upper and lower bounds (details are given in Section 2).…”

“…We prove the following theorem by giving a factor-preserving reduction from Max-2Lin-2 (e.g. see [16]) to Max-PGI k and Max-EGI k .…”