A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

TL;DR

This paper develops a generalized technique for proving lower bounds on polynomial calculus proof degree by leveraging expander properties of clause-variable incidence graphs, and applies it to show certain formulas require high proof complexity.

Contribution

It introduces a clustering-based method to establish polynomial calculus degree lower bounds using expander graphs, extending previous techniques and resolving open questions.

Findings

High PC/PCR degree needed for FPHP formulas on expander graphs

Standard CNF encoding of FPHP requires exponential proof size

Onto-FPHP formulas are easier for polynomial calculus

Abstract

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov '02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.