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The popular matching and condensation problems under matroid constraints

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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.

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Notes

  1. In the conference version, we used the algorithms of Aigner and Dowling (1971), Lawler (1975) for computing a maximum-size common independent set. Furthermore, in this paper, we use the fact that \(|\varPi _X| = O(n)\). Thus, the evaluation in this paper is better than that in the conference version.

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 25730006. The author would like to thank anonymous referees for helpful comments on an earlier version of this paper.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Naoyuki Kamiyama

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  1. Naoyuki Kamiyama
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Correspondence to Naoyuki Kamiyama.

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An earlier version of this paper has appeared in the Proceedings of the 8th Annual International Conference on Combinatorial Optimization and Applications (COCOA), volume 8881 of Lecture Notes in Computer Science, pages 713–728, 2014.

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Kamiyama, N. The popular matching and condensation problems under matroid constraints. J Comb Optim 32, 1305–1326 (2016). https://doi.org/10.1007/s10878-015-9965-8

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  • DOI: https://doi.org/10.1007/s10878-015-9965-8

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