Bounded Unpopularity Matchings
 ChienChung Huang,
 Telikepalli Kavitha,
 Dimitrios Michail,
 Meghana Nasre
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Abstract
We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M′ such that more people prefer M′ to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity—unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NPhard, and that if G does not admit a popular matching, then we have u(M)≥2 for all matchings M in G.
Here we show that a matching M that achieves u(M)=2 can be computed in $O(m\sqrt{n})$ time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H=H _{2},H _{3},…,H _{ k } such that if H _{ k } admits a matching that matches all people, then we can compute in $O(km\sqrt{n})$ time a matching M such that u(M)≤k−1 and $g(M)\le n(1\frac{2}{k})$ . Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.
 Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998) CrossRef
 Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Paretooptimality in house allocation problems. In: Proceedings of 15th Annual International Symposium on Algorithms and Computation. pp. 3–15 (2004)
 Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM J. Comput. 37(4), 1030–1045 (2007)
 Gale, D., Shapley, L.: College admissions and the stability of marriage. Am. Math. Mon. 69, 9–14 (1962) CrossRef
 Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behav. Sci. 20, 166–173 (1975) CrossRef
 Graham, R.L., Grotschel, M., Lovasz, L. (eds.) The Handbook of Combinatorics, vol. 1, pp. 179–232. Elsevier Science, Amsterdam (1995). Chap. 3, Matchings and extensions
 Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)
 Hylland, A., Zeckhauser, R.: The efficient allocation of individuals to positions. J. Polit. Econ. 87(22), 293–314 (1979) CrossRef
 Irving, R.W.: Stable marriage and indifference. Discrete Appl. Math. 48, 261–272 (1994) CrossRef
 Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rankmaximal matchings. ACM Trans. Algorithms 2(4), 602–610 (2006) CrossRef
 Kavitha, T., Mestre, J., Nasre, M.: Popular mixed matchings. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming, pp. 574–584 (2009)
 Kavitha, T., Nasre, M.: Note: optimal popular matchings. Discrete Appl. Math. 157(14), 3181–3186 (2009) CrossRef
 Kavitha, T., Nasre, M.: Popular matchings with variable job capacities. In: Proceedings of 20th Annual International Symposium on Algorithms and Computation, pp. 423–433 (2009)
 Mahdian, M.: Random popular matchings. In: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 238–242 (2006)
 Manlove, D., Sng, C.: Popular matchings in the capacitated house allocation problem. In: Proceedings of the 14th Annual European Symposium on Algorithms, pp. 492–503 (2006)
 McCutchen, R.M.: The leastunpopularityfactor and leastunpopularitymargin criteria for matching problems with onesided preferences. In: Proceedings of the 15th Latin American Symposium on Theoretical Informatics, pp. 593–604 (2008)
 McDermid, E., Irving, R.W.: Popular matchings: Structure and algorithms. In: Proceedings of 15th Annual International Computing and Combinatorics Conference, pp. 506–515 (2009)
 Mestre, J.: Weighted popular matchings. In: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, pp. 715–726 (2006)
 Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. J. Math. Econ. 4, 131–137 (1977) CrossRef
 Yuan, Y.: Residence exchange wanted: a stable residence exchange problem. Eur. J. Oper. Res. 90, 536–546 (1996) CrossRef
 Zhou, L.: On a conjecture by gale about onesided matching problems. J. Econ. Theory 52(1), 123–135 (1990) CrossRef
 Title
 Bounded Unpopularity Matchings
 Journal

Algorithmica
Volume 61, Issue 3 , pp 738757
 Cover Date
 20111101
 DOI
 10.1007/s0045301094349
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Matching with preferences
 Popularity
 Approximation algorithms
 Industry Sectors
 Authors

 ChienChung Huang ^{(1)}
 Telikepalli Kavitha ^{(2)}
 Dimitrios Michail ^{(3)}
 Meghana Nasre ^{(2)}
 Author Affiliations

 1. MaxPlanckInstitut für Informatik, Saarbrücken, Germany
 2. Indian Institute of Science, Bangalore, India
 3. Department of Informatics and Telematics, Harokopion University of Athens, Athens, Greece