多面体的一个不变性原理
An Invariance Principle for Polytopes
摘要：
令X从{-1，1}^n中随机选择，令Y从R^n上的标准球面高斯中随机选择。对于由k个半空间的交所形成的任何（可能无界的）多面体P，我们证明：|Pr [X属于P] - Pr [Y属于P]|< log^{8/5}k * Delta，其中Delta是对于由“规则”半空间（即，影响力低的半空间）。我们的不变性原理的新颖之处在于对k的多对数依赖性。以前，只有至少在k中线性的边界是已知的。我们给出了我们的主要结果的两个重要应用：（1）k个正则半空间的交的布尔噪声敏感性的k中的多对数界（以前的工作给出了k中的线性界）。(2)一个伪随机发生器（PRG），种子长度为O（（log n）*poly（log k，1/delta）），它相对于高斯分布对所有具有k个面的多面体进行delta欺骗。我们也得到PRG具有类似的参数，愚弄多面体的交叉形成的规则半空间的超立方体。使用我们的PRG结构，我们获得了第一个确定性的准多项式时间算法近似计数的解决方案的数量广泛的一类整数规划，包括密集覆盖问题和列联表。
英文摘要：
Let X be randomly chosen from {-1,1}^n, and let Y be randomly chosen from the standard spherical Gaussian on R^n. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr [X belongs to P] - Pr [Y belongs to P]| < log^{8/5}k * Delta, where Delta is a parameter that is small for polytopes formed by the intersection of "regular" halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known. We give two important applications of our main result: (1) A polylogarithmic in k bound on the Boolean noise sensitivity of intersections of k "regular" halfspaces (previous work gave bounds linear in k). (2) A pseudorandom generator (PRG) with seed length O((log n)*poly(log k,1/delta)) that delta-fools all polytopes with k faces with respect to the Gaussian distribution. We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.
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