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

Susanna F. de Rezende; Jakob Nordström; Kilian Risse; Dmitry Sokolov

doi:10.46298/theoretics.25.9

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arXiv · 1912.00534

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

Susanna F. de Rezende·Jakob Nordström·Kilian Risse·Dmitry Sokolov

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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.

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Susanna F. de Rezende, Jakob Nordström, Kilian Risse, Dmitry Sokolov. 2019-12-02. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. https://doi.org/10.46298/theoretics.25.9

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