XNLP-completeness for Parameterized Problems on Graphs with a Linear Structure

TL;DR

This paper establishes XNLP-completeness for several classical graph problems parameterized by linear width measures, indicating their computational hardness and space complexity implications.

Contribution

It proves XNLP-completeness for multiple problems parameterized by linear width measures, strengthening hardness results and suggesting high space requirements for XP algorithms.

Findings

XNLP-hardness for problems parameterized by pathwidth, linear clique-width, and linear mim-width.

Implication that XP algorithms for these problems likely require XP space.

Strengthening of existing W[1]-hardness proofs for these problems.

Abstract

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing $W[1]$-hardness proofs for these problems, since XNLP-hardness implies $W[t]$-hardness for all $t$. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, ($q$-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.