Computing Twin-Width via Treedepth and Vertex Integrity

2026-06-18

AI Summary

This work establishes that determining whether a graph has twin-width at most 4 is NP-hard and demonstrates the absence of fixed-parameter approximation algorithms parameterized solely by twin-width. To overcome this limitation, the paper introduces two novel structural parameters—treedepth and vertex integrity—and presents the first fixed-parameter tractable algorithms for approximating and exactly computing twin-width. Specifically, it develops the first fixed-parameter approximation algorithm for twin-width based on treedepth that does not rely on deletion distance, and provides an optimal contraction sequence via a fixed-parameter exact algorithm parameterized by vertex integrity. A key technical contribution is the introduction of directed twin-width as an intermediate tool, which, combined with the structural properties of the two parameters, enables the design of efficient algorithms.

Abstract

Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width. Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.

Problem

• twin-width
• fixed-parameter tractability
• graph parameter
• NP-hard
• contraction sequences

Innovation

• twin-width
• treedepth
• vertex integrity
• fixed-parameter tractability
• contraction sequences