### 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 +ε for all ε > 0; here αGw ≈ .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(In q)/q^{2}. For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1 - ε)-satisfiable and those which are not even, roughly, (q^{-ε/2})-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [Proceedings of the 38th .Annital 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 - ε vs. ε hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture.

Original language | English (US) |
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Pages (from-to) | 319-357 |

Number of pages | 39 |

Journal | SIAM Journal on Computing |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

### Keywords

- Constraint satisfaction
- Hardness of approximation
- MAX-CUT
- Unique games

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*37*(1), 319-357. https://doi.org/10.1137/S0097539705447372