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.

翻译：摘要：我们研究全局超收缩图（globally hypercontractive graphs）上仿射唯一博弈（affine Unique-Games, UG）的复杂性。这类图并非小集扩张子（small set expanders），但能对所有违反小集扩张性质的小集合提供有用且简洁的特征描述。此类图包括约翰逊图（Johnson graphs）和格拉斯曼图（Grassmann graphs）——它们在近期针对UG的概率可验证明（PCP）构造中起到关键作用——以及通过高维扩张子（high-dimensional expanders）推广的变体。我们的算法展示了如何对平方和（sum-of-squares, SoS）半正定规划的“低熵”解进行舍入，广泛扩展了[BBKSS'21]的算法框架。我们提出了一种新的SoS舍入方案，该方案能消除给定伪分布中的全局相关性，使其在条件化后仍保留多种优良性质。在条件化后获得伪分布的结构控制是许多基于SoS算法的根本性挑战。由于这些挑战，[BBKSS]未能为全局超收缩图建立强健的算法，仅能处理可认证的小集扩张子。我们的成果在多个方面改进了[BBKSS]的结果：可处理任意小（但恒定）的完备性实例；更重要的是，其算法的可靠性保证会随图的其他参数退化（在PCP构造中这些参数均随字母表规模增大），而我们的算法则无此问题。结果表明，全局超收缩图上的UG易于求解，因此凸显了在面向UG的PCP归约背景下，缺乏此类特征描述的图的重要性。