Abstract
The input is a bipartite graph \(G = (\mathcal{A}\cup\mathcal{B}, E)\) where each vertex \(u \in \mathcal{A}\cup\mathcal{B}\) ranks its neighbors in a strict order of preference. A matching M * is said to be popular if there is no matching M such that more vertices are better off in M than in M *. We consider the problem of computing a maximum cardinality popular matching in G. It is known that popular matchings always exist in such an instance G, however the complexity of computing a maximum cardinality popular matching was not known so far. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn 0) algorithm for computing a maximum cardinality popular matching in G, where m = |E| and \(n_0 = \min(|\mathcal{A}|,|\mathcal{B}|)\).
Work done when C.-C. Huang was at MPI Saarbrücken and visited TIFR Mumbai under the IMPECS program.
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Huang, CC., Kavitha, T. (2011). Popular Matchings in the Stable Marriage Problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_56
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DOI: https://doi.org/10.1007/978-3-642-22006-7_56
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