### 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 |
---|---|

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Journal of Combinatorial Optimization |

DOIs | |

Publication status | Accepted/In press - 2018 Apr 27 |

### Fingerprint

### Keywords

- b-matching
- Combinatorial reconfiguration
- Graph algorithm

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Journal of Combinatorial Optimization*, 1-11. https://doi.org/10.1007/s10878-018-0289-3

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, pp. 1-11. https://doi.org/10.1007/s10878-018-0289-3

}

TY - JOUR

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 - 2018/4/27

Y1 - 2018/4/27

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.

KW - b-matching

KW - Combinatorial reconfiguration

KW - Graph algorithm

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U2 - 10.1007/s10878-018-0289-3

DO - 10.1007/s10878-018-0289-3

M3 - Article

SP - 1

EP - 11

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

ER -