Abstract
Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases.
This work was supported by EPSRC grant EP/E011993/1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM Journal on Computing 37, 1030–1045 (2007)
Abraham, D.J., Kavitha, T.: Dynamic matching markets and voting paths. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 65–76. Springer, Heidelberg (2006)
Biró, P., Irving, R.W., Manlove, D.F.: Popular matchings in the Marriage and Roommates problems. Technical Report TR-2009-306, University of Glasgow, Department of Computing Science (December 2009)
Chen, L.: An optimal utility value method with majority equilibrium for public facility allocation. Pacific Journal of Optimization 3(2), 227–234 (2007)
Gabow, H.N.: An efficient implementations of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23(2), 221–234 (1976)
Gabow, H.N., Kaplan, H., Tarjan, R.E.: Unique maximum matching algorithms. Journal of Algorithms 40, 159–183 (2001)
Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences 30, 209–221 (1985)
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph-matching problems. Journal of the ACM 38(4), 815–853 (1991)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)
Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discrete Applied Mathematics 11, 223–232 (1985)
Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Science 20, 166–173 (1975)
Goldberg, A.V.: Scaling algorithms for the shortest paths problem. SIAM Journal on Computing 24(3), 494–504 (1995)
Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)
Huang, C.-C., Kavitha, T., Michail, D., Nasre, M.: Bounded unpopularity matchings. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 127–137. Springer, Heidelberg (2008)
Irving, R.W.: An efficient algorithm for the “stable roommates” problem. Journal of Algorithms 6, 577–595 (1985)
Iwama, K., Manlove, D., Miyazaki, S., Morita, Y.: Stable marriage with incomplete lists and ties. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 443–452. Springer, Heidelberg (1999)
Mahdian, M.: Random popular matchings. In: Proceedings of EC 2006: the 7th ACM Conference on Electronic Commerce, pp. 238–242. ACM, New York (2006)
Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theoretical Computer Science 276(1-2), 261–279 (2002)
Manlove, D.F., Sng, C.T.S.: Popular matchings in the Capacitated House Allocation problem. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 492–503. Springer, Heidelberg (2006)
McCutchen, R.M.: The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 593–604. Springer, Heidelberg (2008)
McDermid, E., Irving, R.W.: Popular matchings: Structure and algorithms. In: Ngo, H.Q. (ed.) COCOON 2009. LNCS, vol. 5609, pp. 506–515. Springer, Heidelberg (2009)
Mestre, J.: Weighted popular matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 715–726. Springer, Heidelberg (2006)
O’Malley, G.: Algorithmic Aspects of Stable Matching Problems. PhD thesis, University of Glasgow, Department of Computing Science (2007)
Ronn, E.: NP-complete stable matching problems. Journal of Algorithms 11, 285–304 (1990)
Roth, A.E., Sotomayor, M.A.O.: Two-sided matching: a study in game-theoretic modeling and analysis. Econometric Society Monographs, vol. 18. Cambridge University Press, Cambridge (1990)
Sng, C.T.S.: Efficient Algorithms for Bipartite Matching Problems with Preferences. PhD thesis, University of Glasgow, Department of Computing Science (2008)
Sng, C.T.S., Manlove, D.F.: Popular matchings in the weighted capacitated house allocation problem. Journal of Discrete Algorithms 8, 102–116 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Biró, P., Irving, R.W., Manlove, D.F. (2010). Popular Matchings in the Marriage and Roommates Problems. In: Calamoneri, T., Diaz, J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13073-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-13073-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13072-4
Online ISBN: 978-3-642-13073-1
eBook Packages: Computer ScienceComputer Science (R0)