### Abstract

Assume that we are given a set of applicants and a set of posts such that each applicant has a preference list over the posts. A matching M between the applicants and the posts is said to be popular if there is no other matching N such that the number of applicants that prefer N to M is larger than the number of applicants that prefer M to N. Then, the goal of the popular matching problem is to decide whether there is a popular matching, and find a popular matching if one exists. Abraham, Irving, Kavitha, and Mehlhorn proved that this problem can be solved in polynomial time even if the preference lists contain ties. In this paper, we consider the popular matching problem with matroid constraints. In this problem, for each post, we are given a matroid on the set of applicants. A set of applicants assigned to each post must be an independent set of its matroid. Kamiyama proved that if there is not a tie in the preference lists, then this problem can be solved in polynomial time. In this paper, we prove that even if there are ties in the preference lists, this problem can be solved in polynomial time.

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

Pages (from-to) | 1801-1819 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Popular matchings with ties and matroid constraints.** / Kamiyama, Naoyuki.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 31, no. 3, pp. 1801-1819. https://doi.org/10.1137/15M104918X

}

TY - JOUR

T1 - Popular matchings with ties and matroid constraints

AU - Kamiyama, Naoyuki

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Assume that we are given a set of applicants and a set of posts such that each applicant has a preference list over the posts. A matching M between the applicants and the posts is said to be popular if there is no other matching N such that the number of applicants that prefer N to M is larger than the number of applicants that prefer M to N. Then, the goal of the popular matching problem is to decide whether there is a popular matching, and find a popular matching if one exists. Abraham, Irving, Kavitha, and Mehlhorn proved that this problem can be solved in polynomial time even if the preference lists contain ties. In this paper, we consider the popular matching problem with matroid constraints. In this problem, for each post, we are given a matroid on the set of applicants. A set of applicants assigned to each post must be an independent set of its matroid. Kamiyama proved that if there is not a tie in the preference lists, then this problem can be solved in polynomial time. In this paper, we prove that even if there are ties in the preference lists, this problem can be solved in polynomial time.

AB - Assume that we are given a set of applicants and a set of posts such that each applicant has a preference list over the posts. A matching M between the applicants and the posts is said to be popular if there is no other matching N such that the number of applicants that prefer N to M is larger than the number of applicants that prefer M to N. Then, the goal of the popular matching problem is to decide whether there is a popular matching, and find a popular matching if one exists. Abraham, Irving, Kavitha, and Mehlhorn proved that this problem can be solved in polynomial time even if the preference lists contain ties. In this paper, we consider the popular matching problem with matroid constraints. In this problem, for each post, we are given a matroid on the set of applicants. A set of applicants assigned to each post must be an independent set of its matroid. Kamiyama proved that if there is not a tie in the preference lists, then this problem can be solved in polynomial time. In this paper, we prove that even if there are ties in the preference lists, this problem can be solved in polynomial time.

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U2 - 10.1137/15M104918X

DO - 10.1137/15M104918X

M3 - Article

VL - 31

SP - 1801

EP - 1819

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -