Counting Popular Matchings in House Allocation Problems
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.
KeywordsBipartite Graph Outgoing Edge Maximum Match Stable Matchings Incoming Edge
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- 2.Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: SODA, pp. 1223–1232 (2008)Google Scholar
- 7.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)Google Scholar
- 9.Lovász, L., Plummer, M.D.: Matching theory. North-Holland Mathematics Studies, vol. 121. North-Holland Publishing Co., Amsterdam (1986), Annals of Discrete Mathematics, 29Google Scholar
- 10.Mahdian, M.: Random popular matchings. In: ACM Conference on Electronic Commerce, pp. 238–242 (2006)Google Scholar
- 13.Nasre, M.: Popular matchings: Structure and cheating strategies. In: STACS, pp. 412–423 (2013)Google Scholar