Abstract
We study the problem of matching applicants to jobs under one-sided preferences; that is, each applicant ranks a non-empty subset of jobs under an order of preference, possibly involving ties. A matching M is said to be more popular than T if the applicants that prefer M to T outnumber those that prefer T to M. A matching is said to be popular if there is no matching more popular than it. Equivalently, a matching M is popular if φ(M,T) ≥ φ(T,M) for all matchings T, where φ(X,Y) is the number of applicants that prefer X to Y.
Previously studied solution concepts based on the popularity criterion are either not guaranteed to exist for every instance (e.g., popular matchings) or are NP-hard to compute (e.g., least unpopular matchings). This paper addresses this issue by considering mixed matchings. A mixed matching is simply a probability distributions over matchings in the input graph. The function φ that compares two matchings generalizes in a natural manner to mixed matchings by taking expectation. A mixed matching P is popular if φ(P,Q) ≥ φ(Q,P) for all mixed matchings Q.
We show that popular mixed matchings always exist and we design polynomial time algorithms for finding them. Then we study their efficiency and give tight bounds on the price of anarchy and price of stability of the popular matching problem.
The second author was supported by an Alexander von Humboldt Fellowship. Part of this work was done when the first author visited MPI für Informatik through the DST-MPG partner group on Efficient Graph Algorithms, IISc Bangalore.
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
Abdulkadiroǧlu, T.S.A.: Ordinal efficiency and dominated sets of assignments. Journal of Economic Theory 112, 157–172 (2003)
Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998)
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM Journal on Computing 37(4), 1030–1045 (2007)
Bogomolnaia, A., Moulin, H.: A new solution to the random assignment problem. Journal of Economic Theory 100(2), 295–328 (2001)
Bogomolnaia, A., Moulin, H.: A simple random assignment problem with a unique solution. Economic Theory 75(3), 257–279 (2004)
Carathéodory, C.: Über den variabilitätsbereich der fourierschen konstanten von positiven harmonischen funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)
Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. jacm 19(2), 248–264 (1972)
Gardenfors, P.: Match making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)
Haung, 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)
Katta, A.-K., Sethuraman, J.: A solution to the random assignment problem on the full preference domain. Journal of Economic Theory 131(1), 231–250 (2005)
Mahdian, M.: Random popular matchings. In: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 238–242 (2006)
Manea, M.: A constructive proof of the ordinal efficiency welfare theorem. Journal of Economic Theory 141, 276–281 (2007)
Manea, M.: Asymptotic ordinal inefficiency of random serial dictatorship. Theoretical Economics (to appear, 2009)
Manlove, D., Sng, C.: 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)
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)
Neumann, J.V.: Zur theorie der gesellschaftsspiele. Math. Annalen 100, 295–320 (1928)
Zhou, L.: On a conjecture by Gale about one-sided matching problems. Journal of Economic Theory 52(1), 123–135 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kavitha, T., Mestre, J., Nasre, M. (2009). Popular Mixed Matchings. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_48
Download citation
DOI: https://doi.org/10.1007/978-3-642-02927-1_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02926-4
Online ISBN: 978-3-642-02927-1
eBook Packages: Computer ScienceComputer Science (R0)