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

Deeparnab Chakrabarty, C. Seshadhri

TL;DR

The paper studies routing in the directed hypercube and its connections to monotonicity testing by reframing Lehman-Ron via flow-cut duality. It first presents an alternate flow-based proof of the LR theorem and then extends it to a generalized dist2 setting, obtaining two edge-disjoint LR solutions when the layer gap satisfies $j-i\ge 2$. It introduces conjectures on flows with simultaneous vertex and edge capacities that would imply robust directed isoperimetric bounds, including the directed Talagrand inequality, thereby linking combinatorial routing to monotonicity testing. The work highlights a path toward purely combinatorial proofs of directed isoperimetry theorems and suggests a rich interaction between routing on the directed hypercube and monotonicity testing theory.

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.

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

TL;DR

The paper studies routing in the directed hypercube and its connections to monotonicity testing by reframing Lehman-Ron via flow-cut duality. It first presents an alternate flow-based proof of the LR theorem and then extends it to a generalized dist2 setting, obtaining two edge-disjoint LR solutions when the layer gap satisfies . It introduces conjectures on flows with simultaneous vertex and edge capacities that would imply robust directed isoperimetric bounds, including the directed Talagrand inequality, thereby linking combinatorial routing to monotonicity testing. The work highlights a path toward purely combinatorial proofs of directed isoperimetry theorems and suggests a rich interaction between routing on the directed hypercube and monotonicity testing theory.

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.
Paper Structure (6 sections, 16 theorems, 2 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Alternate proof of the Lehman-Ron Theorem
  3. Proof of the generalization \ref{['thm:dist2']}
  4. Connections to monotonicity testing
  5. From flows to directed isoperimetry
  6. Connections between conjectures

Key Result

Theorem 1.1

Fix any two integers $i < j$. Let $(S,T;\phi)$ be a matched pair with $S\subseteq L_i$ and $T\subseteq L_j$. Then, there are $|S| = |T|$vertex disjoint paths between $S$ and $T$. We refer to such a set of vertex disjoint paths as an LR solution.

Figures (1)

  • Figure 1: The setup of the proof of \ref{['lem:main']}. The set $X$ is a projection of $A$ along dimension $r$, and similarly $Y$ is a (downward) projection of $B$. The edges between $A$ and $Y$ are projections of edges between $X$ and $B$.

Theorems & Definitions (39)

  • Theorem 1.1: Lehman-Ron Theorem LR01
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof : Proof of \ref{['thm:lr']}
  • ...and 29 more