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On the complexity of minimum maximal acyclic matchings

Author

Listed:
  • Juhi Chaudhary

    (Tata Institute of Fundamental Research)

  • Sounaka Mishra

    (Indian Institute of Technology Madras)

  • B. S. Panda

    (Indian Institute of Technology Delhi)

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.

Suggested Citation

  • Juhi Chaudhary & Sounaka Mishra & B. S. Panda, 2024. "On the complexity of minimum maximal acyclic matchings," Journal of Combinatorial Optimization, Springer, vol. 48(1), pages 1-23, August.
    • Handle: RePEc:spr:jcomop:v:48:y:2024:i:1:d:10.1007_s10878-024-01200-3
    DOI: 10.1007/s10878-024-01200-3
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    References listed on IDEAS

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    1. M. Fürst & D. Rautenbach, 2019. "On some hard and some tractable cases of the maximum acyclic matching problem," Annals of Operations Research, Springer, vol. 279(1), pages 291-300, August.
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