On the complexity of minimum maximal acyclic matchings

Juhi Chaudhary (), Sounaka Mishra () and B. S. Panda ()
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Juhi Chaudhary: Tata Institute of Fundamental Research
Sounaka Mishra: Indian Institute of Technology Madras
B. S. Panda: Indian Institute of Technology Delhi

Journal of Combinatorial Optimization, 2024, vol. 48, issue 1, No 10, 23 pages

Abstract: Abstract Low-Acy-Matching asks to find a maximal matching M in a given graph G of minimum cardinality such that the set of M-saturated vertices induces an acyclic subgraph in G. The decision version of Low-Acy-Matching is known to be $${\textsf{NP}}$$ NP -complete. In this paper, we strengthen this result by proving that the decision version of Low-Acy-Matching remains $${\textsf{NP}}$$ NP -complete for bipartite graphs with maximum degree 6 and planar perfect elimination bipartite graphs. We also show the hardness difference between Low-Acy-Matching and Max-Acy-Matching. Furthermore, we prove that, even for bipartite graphs, Low-Acy-Matching cannot be approximated within a ratio of $$n^{1-\epsilon }$$ n 1 - ϵ for any $$\epsilon >0$$ ϵ > 0 unless $${\textsf{P}}={\textsf{NP}}$$ P = NP . Finally, we establish that Low-Acy-Matching exhibits $$\textsf{APX}$$ APX -hardness when restricted to 4-regular graphs.

Keywords: $$\textsf{NP}$$ NP -completeness; Minimum maximal acyclic matching; $$\textsf{APX}$$ APX -hardness; Acyclic matching (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10878-024-01200-3

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