有向超立方体路由、广义 Lehman-Ron 定理与单调性测试
Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing
AI总结：
本文提出新证明方法以推广 Lehman-Ron 有向超立方体路径定理，并给出同时涉及顶点与边容量的流猜想，连接鲁棒有向 Talagrand 不等式及单调性测试。
摘要：
受单调性测试应用的启发，Lehman 和 Ron（JCTA，2001）证明了在有向超立方体中，可比较的子水平集之间存在一组顶点不相交路径。本文的主要技术贡献是一种新的证明方法，它给出了其定理的一个推广：我们证明存在两个边不相交的顶点不相交路径集合。我们的主要概念贡献是关于同时具有顶点容量和边容量的有向超立方体流的若干猜想，而我们的广义 Lehman-Ron 定理是其特例。我们表明，这些猜想蕴含有向等周定理，尤其是 Khot、Minzer 和 Safra（SIAM J. on Comp，2018）提出的鲁棒有向 Talagrand 不等式。这些等周不等式将超立方体中一个集合的有向表面积与其到单调性的距离联系起来，对于获得布尔函数的最佳单调性测试器一直至关重要。我们相信，我们的猜想为这些有向等周定理的组合证明铺平了道路。
英文摘要：
Motivated by applications to monotonicity testing, Lehman and Ron (JCTA, 2001) proved the existence of a collection of vertex disjoint paths between comparable sub-level sets in the directed hypercube. The main technical contribution of this paper is a new proof method that yields a generalization to their theorem: we prove the existence of two edge-disjoint collections of vertex disjoint paths. Our main conceptual contribution are conjectures on directed hypercube flows with simultaneous vertex and edge capacities of which our generalized Lehman-Ron theorem is a special case. We show that these conjectures imply directed isoperimetric theorems, and in particular, the robust directed Talagrand inequality due to Khot, Minzer, and Safra (SIAM J. on Comp, 2018). These isoperimetric inequalities, that relate the directed surface area (of a set in the hypercube) to its distance to monotonicity, have been crucial in obtaining the best monotonicity testers for Boolean functions. We believe our conjectures pave the way towards combinatorial proofs of these directed isoperimetry theorems.
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