Abstract
We study the problem of counting the number of popular matchings in a given instance. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem.
We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard.
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 J. Comput. 37(4), 1030–1045 (2007)
Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: SODA, pp. 1223–1232 (2008)
Chebolu, P., Goldberg, L.A., Martin, R.A.: The complexity of approximately counting stable matchings. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010. LNCS, vol. 6302, pp. 81–94. Springer, Heidelberg (2010)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The American Mathematical Monthly 69(1), 9–15 (1962)
Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioral Science 20(3), 166–173 (1975)
Irving, R.W., Leather, P.: The complexity of counting stable marriages. SIAM J. Comput. 15(3), 655–667 (1986)
Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In: STOC, pp. 712–721 (2001)
Kavitha, T., Mestre, J., Nasre, M.: Popular mixed matchings. Theor. Comput. Sci. 412(24), 2679–2690 (2011)
Lovász, L., Plummer, M.D.: Matching theory. North-Holland Mathematics Studies, vol. 121. North-Holland Publishing Co., Amsterdam (1986), Annals of Discrete Mathematics, 29
Mahdian, M.: Random popular matchings. In: ACM Conference on Electronic Commerce, pp. 238–242 (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. J. Comb. Optim. 22(3), 339–358 (2011)
Nasre, M.: Popular matchings: Structure and cheating strategies. In: STACS, pp. 412–423 (2013)
Sng, C.T.S., Manlove, D.: Popular matchings in the weighted capacitated house allocation problem. J. Discrete Algorithms 8(2), 102–116 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Acharyya, R., Chakraborty, S., Jha, N. (2014). Counting Popular Matchings in House Allocation Problems. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-06686-8_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06685-1
Online ISBN: 978-3-319-06686-8
eBook Packages: Computer ScienceComputer Science (R0)