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Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists†
Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan
School of Computing Science, University of Glasgow, Glasgow, G12 8QQ, Scotland
Academic Center for Computing and Media Studies, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan
Communication Service Group, Research and Development Center, Nippon Telegraph and Telephone West Corporation, 1-2-31 Sonezaki Kita-ku Osaka 530-0057, Japan
† A part of this paper was presented at the 7th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications (HJ 2011) and the 6th Annual Meeting of the Asian Association for Algorithms and Computation (AAAC 2013).
* Author to whom correspondence should be addressed.
Received: 27 December 2012; in revised form: 17 May 2013 / Accepted: 20 May 2013 / Published: 28 May 2013
Abstract: In the stable marriage problem, any instance admits the so-called man-optimal stable matching, in which every man is assigned the best possible partner. However, there are instances for which all men receive low-ranked partners even in the man-optimal stable matching. In this paper we consider the problem of improving the man-optimal stable matching by changing only one man’s preference list. We show that the optimization variant and the decision variant of this problem can be solved in time O(n3) and O(n2), respectively, where n is the number of men (women) in an input. We further extend the problem so that we are allowed to change k men’s preference lists. We show that the problem is W-hard with respect to the parameter k and give O(n2k+1)-time and O(nk+1)-time exact algorithms for the optimization and decision variants, respectively. Finally, we show that the problems become easy when k = n; we give O(n2.5 log n)-time and O(n2)-time algorithms for the optimization and decision variants, respectively.
Keywords: combinatorial optimization; polynomial-time algorithms; stable marriage problem; man-optimal stable matching
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Inoshita, T.; Irving, R.W.; Iwama, K.; Miyazaki, S.; Nagase, T. Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists. Algorithms 2013, 6, 371-382.
Inoshita T, Irving RW, Iwama K, Miyazaki S, Nagase T. Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists. Algorithms. 2013; 6(2):371-382.
Inoshita, Takao; Irving, Robert W.; Iwama, Kazuo; Miyazaki, Shuichi; Nagase, Takashi. 2013. "Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists." Algorithms 6, no. 2: 371-382.