We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-delta vs. 1-C*delta, for any C> 1, and sufficiently small delta>0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction along with a proof of soundness. The non-unique game we reduce from is similar to non-unique games for which PCP theorems are known. Our proof relies on a new concentration theorem for functions in Gaussian space that are restricted to a random hyperplane. We bound the typical Euclidean distance between the low degree part of the restriction of the function to the hyperplane and the restriction to the hyperplane of the low degree part of the function.
翻译:我们将证明“Boolean United Change Convention Conjecture”(C > 1) 的“Boolean United Change Conjecture”(与1-delta对1-C*delta的偏差为1-C*delta的偏差为1-,而且足够小的delta>0)的问题降低到证明某种非独赢游戏的五氯苯酚理论的问题。在以前的一项工作中,Khot和Moshkovitz建议降低候选人的效率(即,没有健康的证据),目前的工作是首先提供有效的减少,同时提供健康的证据。我们减少的非独赢游戏与已知的五氯苯酚理论的非独角游戏相似。我们的证据依赖于限制在随机超高地的高斯空间功能的新的集中理论。我们把典型的欧克利德距离限制在对高空的功能的低度和低度功能的超高空之间的典型的欧克利德朗距离绑定。