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arXiv · 2205.10995

From Width-Based Model Checking to Width-Based Automated Theorem Proving

Mateus de Oliveira Oliveira·Sam Urmian
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Abstract

In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we prove analytically that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number $k$ as input and correctly determines in time double-exponential in $k^{O(1)}$ whether the conjecture is valid on all graphs of treewidth at most $k$. These upper bounds, which may be regarded as upper-bounds on the size of proofs/disproofs for these conjectures on the class of graphs of treewidth at most $k$, improve significantly on theoretical upper bounds obtained using previously available techniques.

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Mateus de Oliveira Oliveira, Sam Urmian. 2024-09-15. From Width-Based Model Checking to Width-Based Automated Theorem Proving. https://arxiv.org/abs/2205.10995

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