Abstract
We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of the algorithm due to Irving et al. [18] to a non-bipartite setting. Also, we prove several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs.
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., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: EC 2007: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 295–304. ACM Press, New York (2007)
Abraham, D.J., Irving, R.W., Telikepalli, K., Mehlhorn, K.: Popular matchings. In: Proceedings of SODA 2005 the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 424–432. ACM Press, New York (2005)
Abraham, D.J., Levavi, A., Manlove, D.F., O’Malley, G.: The stable roommates problem with globally-ranked pairs. Technical Report TR-2007-257, University of Glasgow, Department of Computing Science (September 2007)
Arkin, E.M., Efrat, A., Mitchell, J.S.B., Polishchuk, V.: Geometric Stable Roommates. Manuscript (2007), http://www.ams.sunysb.edu/~kotya/pages/geomSR.pdf
Bartholdi, J.J., Trick, M.A.: Stable matchings with preferences derived from a psychological model. Operations Research Letters 5, 165–169 (1986)
Chung, K.S.: On the existence of stable roommate matchings. Games and Economic Behavior 33(2), 206–230 (2000)
Delmonico, F.L.: Exchanging kidneys - advances in living-donor transplantation. New England Journal of Medicine 350, 1812–1814 (2004)
Edmonds, J.: Path, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)
Feder, T.: A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences 45, 233–284 (1992)
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)
Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)
Gergely, E.: A simple method for constructing doubly diagonalized latin squares. Journal of Combinatorial Theory, Series A 16(2), 266–272 (1974)
Gjertson, D.W., Cecka, J.M.: Living unrelated donor kidney transplantation. Kidney International 58, 491–499 (2000)
Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)
Irving, R.W.: An efficient algorithm for the stable roommates problem. Journal of Algorithms 6, 577–595 (1985)
Irving, R.W., Manlove, D.F.: The Stable Roommates Problem with Ties. Journal of Algorithms 43, 85–105 (2002)
Irving, R.W., Michail, D., Mehlhorn, K., Paluch, K., Telikepalli, K.: Rank-maximal matchings. ACM Transactions on Algorithms 2(4), 602–610 (2006)
Lovász, L., Plummer, M.D.: Matching Theory. No. 29 in Annals of Discrete Mathematics. North-Holland (1986)
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., O’Malley, G.: Student project allocation with preferences over projects. In: Proceedings of ACiD 2005 the 1st Algorithms and Complexity in Durham workshop. Texts in Algorithmics, vol. 4, pp. 69–80. KCL Publications (2005)
Mehlhorn, K., Michail, D.: Network problems with non-polynomial weights and applications (Unpublished manuscript)
Micali, S., Vazirani, V.V.: An O(|V|. |E|) algorithm for finding maximum matching in general graphs. In: Proceedings of FOCS 1980 the 21st Annual IEEE Symposium on Foundations of Computer Science, pp. 17–27. IEEE Computer Society Press, Los Alamitos (1980)
Ronn, E.: NP-complete stable matching problems. Journal of Algorithms 11, 285–304 (1990)
Roth, A.E., Sönmez, T., Ünver, M.U.: Pairwise kidney exchange. Journal of Economic Theory 125(2), 151–188 (2005)
Roth, A.E., Sönmez, T., Ünver, M.U.: Kidney exchange. Quarterly Journal of Economics 119(2), 457–488 (2004)
Scott, S.: A study of stable marriage problems with ties. PhD thesis, University of Glasgow, Department of Computing Science (2005)
Tan, J.J.M.: A necessary and sufficient condition for the existence of a complete stable matching. J. Algorithms 12(1), 154–178 (1991)
Telikepalli, K., Shah, C.: Efficient algorithms for weighted rank-maximal matchings and related problems. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 153–162. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abraham, D.J., Levavi, A., Manlove, D.F., O’Malley, G. (2007). The Stable Roommates Problem with Globally-Ranked Pairs. In: Deng, X., Graham, F.C. (eds) Internet and Network Economics. WINE 2007. Lecture Notes in Computer Science, vol 4858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77105-0_48
Download citation
DOI: https://doi.org/10.1007/978-3-540-77105-0_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77104-3
Online ISBN: 978-3-540-77105-0
eBook Packages: Computer ScienceComputer Science (R0)