A Uniform Algorithm for Strict NP on Bounded-Treedepth Graphs

2026-10-05

AI Summary

This study addresses the absence of a unified linear-time algorithmic construction and computable parameter-dependence bounds for strict NP problems on graphs of bounded tree-depth. By leveraging tree-depth measurement techniques over Gaifman graphs, this work transforms existential results into a unified algorithmic meta-theorem that achieves linear-time decision and witness construction for given strict NP sentences and relational structures under computable function constraints. The primary contribution is the first unified framework for this setting, which resolves an open problem regarding fixed-parameter tractability and establishes computable complexity bounds. Furthermore, it successfully characterizes the FPT properties of graph parameters such as stack number and queue number, yielding both theoretical advances and efficient decision procedures.

Abstract

A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $\varphi$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\models\varphi$ in time $f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

Problem

• Strict NP
• bounded treedepth
• uniform algorithmic metatheorem
• fixed-parameter tractability
• well-quasi-ordering

Innovation

• bounded treedepth
• Strict NP
• algorithmic metatheorem
• fixed-parameter tractability
• twin-width