A SAT Approach to Twin-Width

TL;DR

This paper introduces a practical SAT-based method for computing the twin-width of graphs, enabling the determination of twin-width for many well-known graphs and identifying minimal graphs for specific bounds.

Contribution

It presents the first efficient SAT encodings for twin-width, facilitating practical computation and analysis of this graph invariant.

Findings

Successfully computed twin-width for many famous graphs

Identified smallest graphs for twin-width bounds 1 to 4

Provided a new tool for analyzing graph complexity

Abstract

The graph invariant twin-width was recently introduced by Bonnet, Kim, Thomass\'e, and Watrigan. Problems expressible in first-order logic, which includes many prominent NP-hard problems, are tractable on graphs of bounded twin-width if a certificate for the twin-width bound is provided as an input. Computing such a certificate, however, is an intrinsic problem, for which no nontrivial algorithm is known. In this paper, we propose the first practical approach for computing the twin-width of graphs together with the corresponding certificate. We propose efficient SAT-encodings that rely on a characterization of twin-width based on elimination sequences. This allows us to determine the twin-width of many famous graphs with previously unknown twin-width. We utilize our encodings to identify the smallest graphs for a given twin-width bound $d \in \{1,\dots,4\}$.