On Parallel-Repetition, Unique-Game and Max-Cut

Khot, Kindler, Mossel and O’Donnell [KKMO04] showed a reduction from the UniqueGame problem to the problem of approximating MaxCut for any approximation that is even slightly better than the efficient algorithm of Goemans & Williamson [GW95]. A most interesting and important open problem is whether there exists an efficient reduction in the other direction. Such a reduction would imply that the complexity of UniqueGame, as well as all other problems shown hard for it, depends on the complexity of approximating MaxCut. The Parallel Repetition technique —or a variation thereof— can yield such a reduction, as long as it is proven to behave ”optimally”, namely so that the upper-bounds for satisfiability match the known lower-bounds. This paper presents two variants of the Parallel-Repetition. One that preserves uniqueness but works only for good expanders and union of disjoint expanders, and the other that works for any instance but does not preserve uniqueness. We show that the two variants of the Parallel-Repetition technique perform “optimally”, i.e, the success probability decays exponentially fast with k, regardless of the alphabet size and with no power on ε, albeit only down to some constant error probability of the generated instance. Such analysis also has algorithmic consequences: it allows converting an approximation algorithm for one set of parameters (error, size of alphabet and approximation ratio) into another, so that an optimal algorithm for one such set of parameters suffices to obtain optimal approximations for all the others. ∗School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. email: safra@post.tau.ac.il †School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. email: odedsc@post.tau.ac.il