Solving Unique Games over Globally Hypercontractive Graphs
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We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. Our algorithm shows how to round "low-entropy" solutions to sum-of-squares (SoS) semidefinite programs, broadly extending the algorithmic framework of [BBKSS'21]. We give a new rounding scheme for SoS, which eliminates global correlations in a given pseudodistribution so that it retains various good properties even after conditioning. Getting structural control over a pseudodistribution after conditioning is a fundamental challenge in many SoS based algorithms. Due to these challenges, [BBKSS] were not able to establish strong algorithms for globally hypercontractive graphs, and could only do so for certifiable small-set expanders. Our results improve upon the results of [BBKSS] in various aspects: we are able to deal with instances with arbitrarily small (but constant) completeness, and most importantly, their algorithm gets a soundness guarantee that degrades with other parameters of the graph (which in all PCP constructions grow with the alphabet size), whereas our doesn't. Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.
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