### 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.

Original language | English |
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Title of host publication | Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings |

Publisher | Springer Verlag |

Pages | 287-296 |

Number of pages | 10 |

Volume | 10392 LNCS |

ISBN (Print) | 9783319623887 |

DOIs | |

Publication status | Published - 2017 |

Event | 23rd International Conference on Computing and Combinatorics, COCOON 2017 - Hong Kong, China Duration: 2017 Aug 3 → 2017 Aug 5 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10392 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 23rd International Conference on Computing and Combinatorics, COCOON 2017 |
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Country | China |

City | Hong Kong |

Period | 17/8/3 → 17/8/5 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings*(Vol. 10392 LNCS, pp. 287-296). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10392 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62389-4_24

**Reconfiguration of maximum-weight b-matchings in a graph.** / Ito, Takehiro; Kakimura, Naonori; Kamiyama, Naoyuki; Kobayashi, Yusuke; Okamoto, Yoshio.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings.*vol. 10392 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10392 LNCS, Springer Verlag, pp. 287-296, 23rd International Conference on Computing and Combinatorics, COCOON 2017, Hong Kong, China, 17/8/3. https://doi.org/10.1007/978-3-319-62389-4_24

}

TY - GEN

T1 - Reconfiguration of maximum-weight b-matchings in a graph

AU - Ito, Takehiro

AU - Kakimura, Naonori

AU - Kamiyama, Naoyuki

AU - Kobayashi, Yusuke

AU - Okamoto, Yoshio

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85028448554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028448554&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-62389-4_24

DO - 10.1007/978-3-319-62389-4_24

M3 - Conference contribution

AN - SCOPUS:85028448554

SN - 9783319623887

VL - 10392 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 287

EP - 296

BT - Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings

PB - Springer Verlag

ER -