Reconfiguration of maximum-weight b-matchings in a graph

Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Yoshio Okamoto

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)1-11
Number of pages11
JournalJournal of Combinatorial Optimization
DOIs
Publication statusAccepted/In press - 2018 Apr 27

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Reconfiguration
Polynomials
Graph in graph theory
Bipartite Graph
Integer
Polynomial time
Vertex of a graph
Integrality
Multiset
Non-negative
Imply

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

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

In: Journal of Combinatorial Optimization, 27.04.2018, p. 1-11.

Research output: Contribution to journalArticle

Ito, Takehiro ; Kakimura, Naonori ; Kamiyama, Naoyuki ; Kobayashi, Yusuke ; Okamoto, Yoshio. / Reconfiguration of maximum-weight b-matchings in a graph. In: Journal of Combinatorial Optimization. 2018 ; pp. 1-11.
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