🤖 AI Summary
This work establishes the first exponential lower bound on the resolution proof length for the Pigeonhole Principle (PHP) and Perfect Matching formulas over sparse, highly unbalanced expander graphs—bridging a long-standing theoretical gap between Ben-Sasson–Wigderson (for balanced constant-degree graphs) and Raz–Razborov (for highly unbalanced dense graphs). Methodologically, it extends Razborov’s pseudo-width technique to the previously intractable setting of highly unbalanced sparse graphs, integrating rigorous expansion analysis with novel combinatorial formula constructions to overcome the original technique’s reliance on graph balance and density. The result significantly broadens the applicability of pseudo-width methods and introduces a new paradigm for analyzing proof complexity over non-uniform graph structures.
📝 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.
