Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing

TL;DR

This paper generalizes Lehman-Ron theorem for directed hypercubes, introduces conjectures on flows with capacities, and connects these to isoperimetric inequalities crucial for monotonicity testing of Boolean functions.

Contribution

It provides a new proof method for a generalized Lehman-Ron theorem and proposes conjectures linking hypercube flows to isoperimetric inequalities.

Findings

Proved existence of two edge-disjoint path collections in directed hypercubes.

Connected hypercube flow conjectures to directed isoperimetric theorems.

Implications for improved monotonicity testing algorithms.

Abstract

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.