[Paper Review] On some hard and some tractable cases of the maximum acyclic matching problem

Maximilian Fürst, Dieter Rautenbach|arXiv (Cornell University)|Oct 23, 2017

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper investigates the computational complexity of the maximum acyclic matching problem, proving that deciding whether a maximum matching is acyclic is NP-complete even for bipartite graphs of maximum degree four. It further shows that maximum weight acyclic matchings can be found efficiently in $P_4$-free and $2P_3$-free graphs, and characterizes graphs in which every maximum matching is acyclic—those whose components are trees or odd cycles.

ABSTRACT

Three well-studied types of subgraph-restricted matchings are induced matchings, uniquely restricted matchings, and acyclic matchings. While it is hard to determine the maximum size of a matching of each of these types, whether some given graph has a maximum matching that is induced or has a maximum matching that is uniquely restricted, can both be decided efficiently. In contrast to that we show that deciding whether a given bipartite graph of maximum degree at most four has a maximum matching that is acyclic is NP-complete. Furthermore, we show that maximum weight acyclic matchings can be determined efficiently for $P_4$-free graphs and $2P_3$-free graphs, and we characterize the graphs for which every maximum matching is acyclic.

Motivation & Objective

• To determine the computational complexity of deciding whether a maximum matching in a graph is acyclic.
• To identify tractable graph classes where the maximum weight acyclic matching problem can be solved efficiently.
• To characterize the family of graphs in which every maximum matching is acyclic.
• To contrast the tractability of acyclic matching with that of induced and uniquely restricted matchings, which are efficiently recognizable when they equal the matching number.

Proposed method

• Reduction from a restricted version of 3-SAT to prove NP-completeness for bipartite graphs of maximum degree four.
• Design of a polynomial-time algorithm for computing maximum weight acyclic matchings in $P_4$-free and $2P_3$-free graphs.
• Structural analysis of graphs where all maximum matchings are acyclic, based on component types (trees and odd cycles).
• Use of graph-theoretic properties: acyclic matchings correspond to subgraphs that are forests; maximum matchings are analyzed via vertex coverage and alternating paths.
• Leveraging known results on line graphs and their squares to explore connections with independent sets and feedback vertex sets.
• Proof by contradiction to show that any cycle in such graphs must be an odd cycle and a connected component.

Experimental results

Research questions

• RQ1 Is it NP-complete to decide whether a maximum matching in a bipartite graph of maximum degree four is acyclic?
• RQ2 Can maximum weight acyclic matchings be computed in polynomial time for $P_4$-free and $2P_3$-free graphs?
• RQ3 What structural properties characterize graphs in which every maximum matching is acyclic?
• RQ4 How does the complexity of deciding $\nu_{ac}(G) = \nu(G)$ compare to that of induced and uniquely restricted matchings?
• RQ5 Is the maximum acyclic matching problem tractable for $P_5$-free graphs, and can techniques from independent feedback vertex sets be applied?

Key findings

• Deciding whether $\nu_{ac}(G) = \nu(G)$ is NP-complete for bipartite graphs of maximum degree at most four.
• Maximum weight acyclic matchings can be computed in polynomial time for $P_4$-free graphs and $2P_3$-free graphs.
• A graph has the property that every maximum matching is acyclic if and only if every component is a tree or an odd cycle.
• The characterization implies that graphs with even cycles or components with high-degree vertices may contain maximum matchings that are not acyclic.
• The result contrasts with the tractability of deciding whether all maximum matchings are induced or uniquely restricted, highlighting a fundamental difference in complexity for acyclic matchings.
• The paper leaves open the complexity of deciding $\nu_{ac}(G) = \nu(G)$ for bipartite graphs of maximum degree three and for $P_5$-free graphs, suggesting potential for future algorithmic advances.