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An Extension of the Birkhoff-von Neumann Theorem to Non-Bipartite Graphs

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  • Vijay V. Vazirani

Abstract

We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively "removes" from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriately saves the day.

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  • Vijay V. Vazirani, 2020. "An Extension of the Birkhoff-von Neumann Theorem to Non-Bipartite Graphs," Papers 2010.05984, arXiv.org, revised Oct 2020.
    • Handle: RePEc:arx:papers:2010.05984
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    References listed on IDEAS

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    1. Roth, Alvin E. & Sonmez, Tayfun & Utku Unver, M., 2005. "Pairwise kidney exchange," Journal of Economic Theory, Elsevier, vol. 125(2), pages 151-188, December.
    2. Hylland, Aanund & Zeckhauser, Richard, 1979. "The Efficient Allocation of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 87(2), pages 293-314, April.
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    1. Ioannis Panageas & Thorben Trobst & Vijay V. Vazirani, 2021. "Time-Efficient Algorithms for Nash-Bargaining-Based Matching Market Models," Papers 2106.02024, arXiv.org, revised Nov 2024.

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