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Maximum Matchings and Short Voting Paths

Author Telikepalli Kavitha

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LIPIcs.MFCS.2026.56.pdf
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  • 18 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Bipartite graphs
  • Fractional matchings
  • Polytopes
  • LP duality

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Abstract

Our input is a marriage instance G = (A ∪ B, E), i.e., it is a bipartite graph where every vertex has strict preferences over its neighbors. The preferences that a vertex has on its neighbors extend naturally to preferences over matchings. A maximum matching M that does not lose an election against any maximum matching (where vertices cast votes) is a popular maximum-matching. These matchings are useful in practice - they always exist and can be efficiently computed [Kavitha, SICOMP 2014]. Suppose preferences change; then the problem is to update the current matching M_0 via a short voting path to a popular maximum-matching, where a length-𝓁 voting path from M_0 to M_𝓁 is a sequence of maximum matchings ⟨M_0,M_1,…,M_𝓁⟩ such that each matching is more popular than its predecessor. There are maximum matchings from which there is no voting path (of any length) to a popular maximum-matching [Bhattacharya et al., ICALP 2015]. 
We show a polynomial-time algorithm to decide if there exists a short voting path, i.e. one of length ≤ 2, from a given maximum matching to a popular maximum-matching and find one, if so. Voting paths motivate natural relaxations of popularity: pseudo-popular maximum-matchings and mostly-popular maximum-matchings; these yield more egalitarian or optimal solutions than popular maximum-matchings. We show polynomial-time algorithms to compute such optimal solutions that go beyond popularity. In particular, we give a combinatorial characterization of pseudo-popular maximum-matchings in terms of forced vertices and forbidden edges. We also show a polynomial-time algorithm to compute at most |E| = m popular maximum-matchings such that any maximum matching that loses to some popular maximum-matching loses to at least one of these m matchings.

Cite As Get BibTex

Telikepalli Kavitha. Maximum Matchings and Short Voting Paths. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.56

Author Details

Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

Supported by the Department of Atomic Energy, Government of India, under project no. RTI4014.

Acknowledgements

Thanks to the reviewers of this paper for their helpful comments and suggestions that improved the presentation of the paper.

References

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