CIAC 2013: Algorithms and Complexity pp 324-335

# Capacitated Rank-Maximal Matchings

• Katarzyna Paluch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7878)

## 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.

## Keywords

Bipartite Graph Matchings Problem Maximal Rank Good Path Combinatorial Algorithm
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
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)
2. 2.
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)
3. 3.
Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular Matchings. SIAM J. Comput. 37(4), 1030–1045 (2007)
4. 4.
Huang, C.-C., Kavitha, T., Michail, D., Nasre, M.: Bounded Unpopularity Matchings. Algorithmica 61(3), 738–757 (2011)
5. 5.
Gabow, H.N.: An Efficient Reduction Technique for Degree-Constrained Subgraph and Bidirected Network Flow Problems STOC, pp. 448–456 (1983)Google Scholar
6. 6.
Irving, R.W.: Greedy matchings. Technical report TR-2003-136, University of Glasgow (April 2003)Google Scholar
7. 7.
Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.E.: Rank-maximal matchings. ACM Transactions on Algorithms 2(4), 602–610 (2006)
8. 8.
Lovasz, L., Plummer, M.D.: Matching Theory. Ann. Discrete Math., vol. 29. North-Holland, Amsterdam (1986)Google Scholar
9. 9.
Mahdian, M.: Random popular matchings. In: ACM Conference on Electronic Commerce, pp. 238–242 (2006)Google Scholar
10. 10.
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)
11. 11.
Mehlhorn, K., Michail, D.: Network Problems with Non-Polynomial Weights and Applications (2005) (manuscript)Google Scholar
12. 12.
Michail, D.: Reducing rank-maximal to maximum weight matching. Theor. Comput. Sci. 389(1-2), 125–132 (2007)
13. 13.
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)
14. 14.
Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. J. Math. Econom. 4, 536–546 (1977)