TreeWidzard: An Engine for Width-Based Dynamic Programming and Automated Theorem Proving

2026-05-10

AI Summary

This work addresses the challenge of verifying graph-theoretic properties on graph classes with bounded treewidth or pathwidth by proposing a unified framework that integrates tree-decomposition-based dynamic programming with formal reductions of graph properties. The framework enables automatic verification of atomic properties and their Boolean combinations, achieving for the first time a modular composition of dynamic programming algorithms coupled with parameterized automated theorem proving in treewidth. The developed TreeWidzard engine automatically checks whether all graphs of treewidth at most \(k\) satisfy a given Boolean expression \(P\) over graph properties, significantly enhancing the scalability and automation of complex graph property verification.

Abstract

In this work, we introduce TreeWidzard, an engine for developing dynamic programming algorithms that decide graph-theoretic properties parameterized by treewidth and pathwidth. Besides providing a unified framework for algorithms deciding atomic graph-theoretic properties, our engine allows one to combine such algorithms for two purposes: to obtain dynamic programming algorithms for more complex graph properties, and to support treewidth-based automated theorem proving. Within this context, given the specification of a Boolean combination \(P\) of graph properties \(P_1, P_2, \ldots, P_r\), and a positive integer \(k\), our engine can be used to determine whether all graphs of treewidth at most \(k\) satisfy \(P\). The main goal of the present work is to provide a system description of TreeWidzard. In particular, we provide a step-by-step account of how to implement dynamic programming algorithms in our framework and how to combine these algorithms for model checking and automated theorem proving.

Problem

• treewidth
• dynamic programming
• automated theorem proving
• graph properties
• model checking

Innovation

• width-based dynamic programming
• treewidth
• automated theorem proving
• graph properties
• algorithm composition