Short Resolution Refutations for CNFs with Bounded Weighted Incidence Treewidth

2026-10-01

AI Summary

This study addresses the open problem of whether CNF formulas admit fixed-parameter tractable (FPT)-size resolution refutations when parameterized by incidence treewidth. Methodologically, it introduces novel variants such as (partial) logarithmically weighted incidence treewidth, integrating tree decompositions with weighted graph-theoretic techniques to construct k-DNF refutations and efficiently translate them into resolution refutations. The core contribution lies in establishing, for the first time, the existence of FPT-size and regular resolution refutations under these new parameters. Furthermore, this work derives polynomial upper bounds that significantly tighten the theoretical limits on both refutation length and width, thereby advancing the understanding of proof complexity within parameterized frameworks.

Abstract

It is an open problem in proof complexity whether every unsatisfiable CNF formula has an FPT-sized resolution refutation parameterized by incidence treewidth. In this paper, we establish several upper bounds on resolution refutation length related to this problem. Consider an unsatisfiable CNF formula $F$ with $n$ variables, $m$ clauses, maximum clause width $k$, and incidence treewidth $\mathrm{tw}^*(F)$. In this paper, we introduce two variants of incidence treewidth. Their definitions can be stated informally as follows. The first is log-weighted incidence treewidth $\mathrm{tw}_{\log}^*(F)$, which is the treewidth of the weighted incidence graph, in which variables have weight one and each clause has weight equal to the logarithm of its width. The second is partially log-weighted incidence treewidth $\mathrm{tw}^*_{\mathrm{plog}}(F)$, which is a refinement of log-weighted incidence treewidth. In this variant, for a nice tree decomposition of the incidence graph, each clause has weight one along a path selected for that clause and elsewhere has weight equal to the logarithm of one plus the number of its literals whose variables do not appear in any bag on that path, and variables have weight one. For every unsatisfiable CNF formula $F$, we prove the existence of (i) an FPT-sized resolution refutation parameterized by log-weighted incidence treewidth, with width at most $\mathrm{tw}_{\log}^*(F)+k$; (ii) a resolution refutation of length $(n+m)k^{O(\mathrm{tw}^*(F))}$ and width at most $\mathrm{tw}^*(F)+k$; (iii) an FPT-sized resolution refutation parameterized by partially log-weighted incidence treewidth; and (iv) an FPT-sized regular resolution refutation parameterized by log-weighted incidence treewidth. Our main idea is to construct FPT-sized $k$-DNF resolution refutations parameterized by incidence treewidth, and then convert them into resolution refutations.

Problem

• proof complexity
• resolution refutation
• CNF formula
• incidence treewidth
• fixed-parameter tractable

Innovation

• Resolution Refutation
• Incidence Treewidth
• Log-weighted Treewidth
• Proof Complexity
• k-DNF Resolution