The Popular Matching and Condensation Problems Under Matroid Constraints

Kamiyama, Naoyuki

doi:10.1007/978-3-319-12691-3_53

Cited by 2 publications

(3 citation statements)

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“…For each post p in P , we are given a matroid M p = (E(p), I p ). We assume that for every edge (a, p) in E, {(a, p)} is an independent set of M p (see [8] for concrete examples of matroid constraints). As in [1], we assume that for each applicant a in A, there exist last resort posts ℓ 1 (a), ℓ 2 (a) in P satisfying the following conditions.…”

Section: Problem Formulationmentioning

confidence: 99%

“…For example, Kamiyama [8,9] proposed polynomial-time algorithms for a matroid generalization of the (ordinary) popular matching problem. A matroid constraint is a generalization of several capacity constraints (see, e.g., [8]). …”

Section: Introductionmentioning

confidence: 99%

“…Abraham, Irving, Kavitha, and Mehlhorn [1] presented polynomial-time algorithms for the problem of deciding whether there exists a popular matching, and finding a popular matching if one exists. Since the seminal paper by Abraham, Irving, Kavitha, and Mehlhorn [1], several variants of the popular matching problem [8,9,15,17,21] and related problems [8,10]- [13,16,22,23] have been extensively investigated. For example, Manlove and Sng [15] proposed polynomial-time algorithms for a many-to-one variant of the popular matching problem.…”

Section: Introductionmentioning

confidence: 99%

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A Characterization of Weighted Popular Matchings Under Matroid Constraints

Kamiyama¹

2018

JORSJ

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The popular matching problem introduced by Abraham, Irving, Kavitha, and Mehlhorn is one of bipartite matching problems with one-sided preference lists. In this paper, we first propose a matroid generalization of the weighted variant of popular matchings introduced by Mestre. Then we give a characterization of weighted popular matchings in bipartite graphs with matroid constraints and one-sided preference lists containing no ties. This characterization is based on the characterization of weighted popular matchings proved by Mestre. Lastly we prove that we can decide whether a given matching is a weighted popular matching under matroid constraints in polynomial time by using our characterization.Keywords: Discrete optimization, popular matching, matroid IntroductionThe popular matching problem introduced by Abraham, Irving, Kavitha, and Mehlhorn [1] is one of assignment problems in bipartite graphs with one-sided preference lists. Roughly speaking, a matching M is said to be popular, if there exists no other matching N such that the number of agents that prefer N to M is larger than the number of agents that prefer M to N . The concept of popularity was originally proposed by Gärdenfors [5] proposed polynomial-time algorithms for a many-to-one variant of the popular matching problem. See, e.g, [6] for the application of the popular matching problem to real-world problems.In this paper, we focus on the weighted variant of the popular matching problem introduced by Mestre [17]. In the (ordinary) popular matching problem, we consider the number of agents that prefer some matching to another matching. In other words, the opinion of every agent is valued equally. On the other hand, in this weighted variant, the opinions of agents are not valued equally. More precisely, in this setting, each agent has a weight. A matching M is said to be popular, if there exists no other matching N such that the sum of the weights of agents that prefer N to M is larger than the sum of the weights of agents that prefer M to N . Mestre [17] gave polynomial-time algorithms for the problem of deciding whether there exists a weighted popular matching, and finding a weighted popular matching if one exists. Furthermore, Sng and Manlove [21] proved that if the preference lists do not contain ties, then this problem in a many-to-one setting can be solved in polynomial time.

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Section: Problem Formulationmentioning

confidence: 99%