Abstract
We consider capacitated rank-maximal matchings. Rank-maximal matchings have been considered before and are defined as follows. We are given a bipartite graph \( G= (\mathcal{A} \cup \mathcal{P}, {\cal E})\), in which \(\mathcal{A}\) denotes applicants, \(\mathcal{P}\) posts and edges have ranks – an edge (a,p) has rank i if p belongs to (one of) a’s ith choices. A matching M is called rank-maximal if the largest possible number of applicants is matched in M to their first choice posts and subject to this condition the largest number of appplicants is matched to their second choice posts and so on. We give a combinatorial algorithm for the capacitated version of the rank-maximal matching problem, in which each applicant or post v has capacity b(v). The algorithm runs in \(O(\min(B,C \sqrt{B} ) m)\) time, where C is the maximal rank of an edge in an optimal solution and \(B= \min (\sum_{a \in \mathcal{A}} {b(a)}, \sum_{p \in \mathcal{P}}{b(p)})\) and n, m denote the number of vertices/edges respectively. (B depends on the graph, however it never exceeds m.) The previously known algorithm [11] for this problem has a worse running time of O(Cnmlog(n 2/m) logn) and is not combinatorial –it is based on a weakly polynomial algorithm of Gabow and Tarjan using scaling. To construct the algorithm we use the generalized Gallai-Edmonds decomposition theorem, which we prove in a convenient form for our purposes. As a by-product we obtain a faster (by a factor of \(O(\sqrt{n})\)) algorithm for the Capacitated House Allocation with Ties problem.
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References
Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto Optimality in House Allocation Problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 3–15. Springer, Heidelberg (2004)
Abraham, D.J., Chen, N., Kumar, V., Mirrokni, V.S.: Assignment Problems in Rental Markets. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 198–213. Springer, Heidelberg (2006)
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular Matchings. SIAM J. Comput. 37(4), 1030–1045 (2007)
Huang, C.-C., Kavitha, T., Michail, D., Nasre, M.: Bounded Unpopularity Matchings. Algorithmica 61(3), 738–757 (2011)
Gabow, H.N.: An Efficient Reduction Technique for Degree-Constrained Subgraph and Bidirected Network Flow Problems STOC, pp. 448–456 (1983)
Irving, R.W.: Greedy matchings. Technical report TR-2003-136, University of Glasgow (April 2003)
Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.E.: Rank-maximal matchings. ACM Transactions on Algorithms 2(4), 602–610 (2006)
Lovasz, L., Plummer, M.D.: Matching Theory. Ann. Discrete Math., vol. 29. North-Holland, Amsterdam (1986)
Mahdian, M.: Random popular matchings. In: ACM Conference on Electronic Commerce, pp. 238–242 (2006)
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)
Mehlhorn, K., Michail, D.: Network Problems with Non-Polynomial Weights and Applications (2005) (manuscript)
Michail, D.: Reducing rank-maximal to maximum weight matching. Theor. Comput. Sci. 389(1-2), 125–132 (2007)
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)
Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. J. Math. Econom. 4, 536–546 (1977)
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Paluch, K. (2013). Capacitated Rank-Maximal Matchings. In: Spirakis, P.G., Serna, M. (eds) Algorithms and Complexity. CIAC 2013. Lecture Notes in Computer Science, vol 7878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38233-8_27
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DOI: https://doi.org/10.1007/978-3-642-38233-8_27
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