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تسجيل مستخدم جديدStatistical physics approaches to Unique Games
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We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the yes case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the yes case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would refute the Unique Games Conjecture.
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In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Un
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Covering spaces of graphs have long been useful for studying expanders (as graph lifts) and unique games (as the label-extended graph). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology a
nd the Unique Games Conjecture. Our starting point is Linials 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-ODonnell, FOCS 2004; SICOMP, 2007) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new Unique Games-complete problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over $mathbb{Z}/2mathbb{Z}$, and we show Unique Games-completeness over $mathbb{Z}/kmathbb{Z}$ for large $k$.) This equivalence comes from the fact that when the structure group $G$ of the covering space is Abelian - or more generally for principal $G$-bundles - Maximum Section of a $G$-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.
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