Structural Parameterizations for Induced and Acyclic Matching

TL;DR

This paper refines the complexity bounds for Induced and Acyclic Matching problems parameterized by treewidth and clique-width, establishing optimal algorithms and lower bounds under the Strong Exponential Time Hypothesis.

Contribution

It provides tight bounds and optimal algorithms for these problems under various structural parameters, resolving open questions from prior work.

Findings

A more precise analysis shows Acyclic Matching algorithm runs in 5^{tw}n^{O(1)} time.

Lower bounds under pw-SETH match the upper bounds, confirming optimality.

Single-exponential FPT algorithms are developed for clique-width, with proven optimality.

Abstract

We revisit the (structurally) parameterized complexity of Induced Matching and Acyclic Matching, two problems where we seek to find a maximum independent set of edges whose endpoints induce, respectively, a matching and a forest. Chaudhary and Zehavi [WG '23] recently studied these problems parameterized by treewidth, denoted by $\mathrm{tw}$. We resolve several of the problems left open in their work and extend their results as follows: (i) for Acyclic Matching, Chaudhary and Zehavi gave an algorithm of running time $6^{\mathrm{tw}}n^{\mathcal{O}(1)}$ and a lower bound of $(3-\varepsilon)^{\mathrm{tw}}n^{\mathcal{O}(1)}$ (under the SETH); we close this gap by, on the one hand giving a more careful analysis of their algorithm showing that its complexity is actually $5^{\mathrm{tw}} n^{\mathcal{O}(1)}$, and on the other giving a pw-SETH-based lower bound showing that this running time cannot be improved (even for pathwidth), (ii) for Induced Matching we show that their $3^{\mathrm{tw}} n^{\mathcal{O}(1)}$ algorithm is optimal under the pw-SETH (in fact improving over this for pathwidth or even for cutwidth is equivalent to falsifying the pw-SETH) by adapting a recent reduction for Bounded Degree Vertex Deletion, (iii) for both problems we give FPT algorithms with single-exponential dependence when parameterized by clique-width and in particular for Induced Matching our algorithm has running time $3^{\mathrm{cw}} n^{\mathcal{O}(1)}$, which is optimal under the pw-SETH from our previous result.