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The popular matching and condensation problems under matroid constraints

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  • Naoyuki Kamiyama

    (Kyushu University)

Abstract

The popular matching problem introduced by Abraham, Irving, Kavitha, and Mehlhorn is a matching problem in which there exist applicants and posts, and applicants have preference lists over posts. A matching M is said to be popular, if there exists 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. The goal of this problem is to decide whether there exists a popular matching, and find a popular matching if one exists. In this paper, we first consider a matroid generalization of the popular matching problem with strict preference lists, and give a polynomial-time algorithm for this problem. In the second half of this paper, we consider the problem of transforming a given instance of a matroid generalization of the popular matching problem with strict preference lists by deleting a minimum number of applicants so that it has a popular matching. This problem is a matroid generalization of the popular condensation problem with strict preference lists introduced by Wu, Lin, Wang, and Chao. By using the results in the first half, we give a polynomial-time algorithm for this problem.

Suggested Citation

  • Naoyuki Kamiyama, 2016. "The popular matching and condensation problems under matroid constraints," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1305-1326, November.
    • Handle: RePEc:spr:jcomop:v:32:y:2016:i:4:d:10.1007_s10878-015-9965-8
    DOI: 10.1007/s10878-015-9965-8
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    References listed on IDEAS

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    1. Telikepalli Kavitha & Meghana Nasre & Prajakta Nimbhorkar, 2014. "Popularity at minimum cost," Journal of Combinatorial Optimization, Springer, vol. 27(3), pages 574-596, April.
    2. Satoru Fujishige & Akihisa Tamura, 2007. "A Two-Sided Discrete-Concave Market with Possibly Bounded Side Payments: An Approach by Discrete Convex Analysis," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 136-155, February.
    3. Eric McDermid & Robert W. Irving, 2011. "Popular matchings: structure and algorithms," Journal of Combinatorial Optimization, Springer, vol. 22(3), pages 339-358, October.
    4. Tamás Fleiner, 2003. "A Fixed-Point Approach to Stable Matchings and Some Applications," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 103-126, February.
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