Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

TL;DR

This paper establishes exponential lower bounds on resolution proof lengths for pigeonhole principle and perfect matching formulas over sparse expander graphs, advancing understanding of proof complexity in this regime.

Contribution

It extends Razborov's pseudo-width method to sparse graphs, providing new exponential lower bounds and demonstrating the method's broader applicability.

Findings

Proves exponential lower bounds for PHP formulas over sparse graphs.

Extends pseudo-width method to new graph regimes.

Highlights potential for broader application of the method.

Abstract

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.