Structural Parameterizations for Induced and Acyclic Matching

2025-02-20

AI Summary

This paper investigates the exact computational complexity of Induced Matching and Acyclic Matching under structural graph parameters—treewidth (tw), pathwidth (pw), and clique-width (cw). Using dynamic programming, state compression, pw-SETH-based reductions, and clique-width decompositions, we establish tight complexity bounds. First, we obtain the optimal time bound $3^{ ext{tw}} cdot n^{O(1)}$ for Induced Matching, resolving its precise treewidth dependence. Second, we improve the treewidth-based upper bound for Acyclic Matching from $6^{ ext{tw}}$ to $5^{ ext{tw}}$ and prove its tightness. Third, we design the first single-exponential FPT algorithm for Acyclic Matching parameterized by clique-width, running in $3^{ ext{cw}} cdot n^{O(1)}$ time, and show its optimality assuming SETH. All results match current theoretical lower bounds, thereby closing several long-standing gaps in the parameterized complexity of matching variants.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

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 is emph{equivalent} to falsifying the pw-SETH) by adapting a recent reduction for extsc{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 extsc{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.

Problem

• Optimize Acyclic Matching algorithm complexity.
• Prove Induced Matching algorithm optimality.
• Develop FPT algorithms for clique-width parameterization.

Innovation

• Optimized treewidth parameterized algorithms
• Enhanced lower bounds under SETH
• FPT algorithms for clique-width