### Abstract

Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F. This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-matroid matching problem, where the given delta-matroid is a projection of a linear delta-matroid. We first show that it can be solved in polynomial time if the given linear delta-matroid is generic. This result enlarges a polynomially solvable class of matching problems with precedence constraints on vertices such as the 2-master/slave matching. In addition, we design a polynomial-time algorithm when the graph is bipartite and the delta-matroid is defined on one vertex side. This result is extended to the case where a linear matroid constraint is additionally imposed on the other vertex side.

Original language | English |
---|---|

Pages (from-to) | 942-961 |

Number of pages | 20 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Constrained matching
- Delta-matroid
- Mixed matrix theory
- Polynomial-time algorithm

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*28*(2), 942-961. https://doi.org/10.1137/110860070

**Matching problems with delta-matroid constraints.** / Kakimura, Naonori; Takamatsu, Mizuyo.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 28, no. 2, pp. 942-961. https://doi.org/10.1137/110860070

}

TY - JOUR

T1 - Matching problems with delta-matroid constraints

AU - Kakimura, Naonori

AU - Takamatsu, Mizuyo

PY - 2014

Y1 - 2014

N2 - Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F. This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-matroid matching problem, where the given delta-matroid is a projection of a linear delta-matroid. We first show that it can be solved in polynomial time if the given linear delta-matroid is generic. This result enlarges a polynomially solvable class of matching problems with precedence constraints on vertices such as the 2-master/slave matching. In addition, we design a polynomial-time algorithm when the graph is bipartite and the delta-matroid is defined on one vertex side. This result is extended to the case where a linear matroid constraint is additionally imposed on the other vertex side.

AB - Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F. This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-matroid matching problem, where the given delta-matroid is a projection of a linear delta-matroid. We first show that it can be solved in polynomial time if the given linear delta-matroid is generic. This result enlarges a polynomially solvable class of matching problems with precedence constraints on vertices such as the 2-master/slave matching. In addition, we design a polynomial-time algorithm when the graph is bipartite and the delta-matroid is defined on one vertex side. This result is extended to the case where a linear matroid constraint is additionally imposed on the other vertex side.

KW - Constrained matching

KW - Delta-matroid

KW - Mixed matrix theory

KW - Polynomial-time algorithm

UR - http://www.scopus.com/inward/record.url?scp=84904007381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904007381&partnerID=8YFLogxK

U2 - 10.1137/110860070

DO - 10.1137/110860070

M3 - Article

VL - 28

SP - 942

EP - 961

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -