State Canonization and Early Pruning in Width-Based Automated Theorem Proving

TL;DR

This paper advances width-based automated theorem proving by introducing state-canonization and early-pruning techniques, demonstrating practical applications in graph theory conjectures, including verifying Reed's conjecture on specific graph classes.

Contribution

It introduces novel state-canonization and early-pruning methods to improve the efficiency of width-based theorem proving in graph theory.

Findings

Reed's conjecture verified for graphs of pathwidth at most 5 and treewidth at most 3.

The framework can automatically find counterexamples to invalid conjecture strengthenings.

The proposed techniques significantly reduce the number of states evaluated during theorem proving.

Abstract

Width-based automated theorem proving is a framework where counterexamples to graph-theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time \(2^{2^{k^{O(1)}}}\) on the class of graphs of treewidth at most \(k\). In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counterexample of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form \(\mathcal{P}_1 \rightarrow \mathcal{P}_2\), for graph properties \(\mathcal{P}_1\) and \(\mathcal{P}_2\), where \(\mathcal{P}_1\) is a property closed under subgraphs. As a concrete application, we use our framework in the study of graph-theoretic conjectures related to coloring triangle-free graphs. In particular, our algorithm is able to show that Reed's conjecture for triangle-free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counterexamples to invalid strengthenings of Reed's conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.