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International Computing and Combinatorics Conference

COCOON 2017: Computing and Combinatorics pp 287–296Cite as

Reconfiguration of Maximum-Weight b-Matchings in a Graph

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10392))

Abstract

Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs.

T. Ito – Supported by JST CREST Grant Number JPMJCR1402, Japan, and JSPS KAKENHI Grant Number JP16K00004.

N. Kakimura – Supported by JST ERATO Grant Number JPMJER1305, Japan, and by JSPS KAKENHI Grant Number JP17K00028.

N. Kamiyama – Supported by JST PRESTO Grant Number JPMJPR14E1, Japan.

Y. Kobayashi – Supported by JST ERATO Grant Number JPMJER1305, Japan, and by JSPS KAKENHI Grant Numbers JP16K16010 and JP16H03118.

Y. Okamoto – Supported by Kayamori Foundation of Informational Science Advancement, JST CREST Grant Number JPMJCR1402, Japan, and JSPS KAKENHI Grant Numbers JP24106005, JP24700008, JP24220003, JP15K00009.

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Notes

  1. 1.

    Properly speaking, both Ito et al. [11] and M\(\mathrm{\ddot{u}}\)hlenthaler [20] studied their reconfiguration problems under a more generalized reconfiguration rule, called the TAR (Token Addition and Removal) rule. Their results hold also under the reconfiguration rule of this paper, which is called the TJ (Token Jumping) rule.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Takehiro Ito

  2. Department of Mathematics, Keio University, Yokohama, Japan

    Naonori Kakimura

  3. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Naoyuki Kamiyama

  4. Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan

    Yusuke Kobayashi

  5. Department of Communication Engineering and Informatics, University of Electro-Communications, Chofu, Japan

    Yoshio Okamoto

Authors
  1. Takehiro Ito
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  2. Naonori Kakimura
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  3. Naoyuki Kamiyama
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  4. Yusuke Kobayashi
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  5. Yoshio Okamoto
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Correspondence to Takehiro Ito .

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Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

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Ito, T., Kakimura, N., Kamiyama, N., Kobayashi, Y., Okamoto, Y. (2017). Reconfiguration of Maximum-Weight b-Matchings in a Graph. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_24

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  • DOI: https://doi.org/10.1007/978-3-319-62389-4_24

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  • Print ISBN: 978-3-319-62388-7

  • Online ISBN: 978-3-319-62389-4

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