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Excluded t-Factors in Bipartite Graphs: A Unified Framework for Nonbipartite Matchings and Restricted 2-Matchings

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Integer Programming and Combinatorial Optimization (IPCO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10328))

Abstract

We propose a new framework of optimal t-matchings excluding prescribed t-factors in bipartite graphs. It is a generalization of the nonbipartite matching problem and includes a number of generalizations such as the triangle-free 2-matching, square-free 2-matching, and even factor problems. We demonstrate a unified understanding of those generalizations by designing a combinatorial algorithm for our problem under a reasonable assumption, which is broad enough to include the specific problems listed above. We first present a min-max theorem and a combinatorial algorithm for the unweighted version. We further provide a linear programming formulation with dual integrality and a primal-dual algorithm for the weighted version. A key ingredient of our algorithm is a technique of shrinking forbidden structures, which commonly extends the techniques of shrinking odd cycles, triangles, and squares in Edmonds’ blossom algorithm, in the triangle-free 2-matching algorithm, and in the square-free 2-matching algorithm, respectively.

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Acknowledgements

The author is obliged to Yutaro Yamaguchi for the helpful comments on the draft of the paper. The author is also thankful to anonymous referees for their careful reading and comments. This research is partially supported by JSPS KAKENHI Grant Number 16K16012.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Faculty of Science and Engineering, Hosei University, Tokyo, 184-8584, Japan

    Kenjiro Takazawa

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  1. Kenjiro Takazawa
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Correspondence to Kenjiro Takazawa .

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Editors and Affiliations

  1. Ecole Polytechnique Federale de Lausanne - EPFL, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. University of Waterloo, Waterloo, Ontario, Canada

    Jochen Koenemann

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Takazawa, K. (2017). Excluded t-Factors in Bipartite Graphs: A Unified Framework for Nonbipartite Matchings and Restricted 2-Matchings. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_35

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59249-7

  • Online ISBN: 978-3-319-59250-3

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