Abstract
Many real-life optimization problems are naturally formulated as questions about matchings in (bipartite) graphs.
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We have a bipartite graph. The edge set is partitioned into classes E 1, E 2, . . . , E r . For a matching M, let s i be the number of edges in M ∩ E i . A rank − maximal matching maximizes the vector (s 1, s 2, . . . , s r ). We show how to compute a rank-maximal matching in time O(r\(\sqrt{nm}\)) [IKM + 06].
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We have a bipartite graph. The vertices on one side of the graph rank the vertices on the other side; there are no ties. We call a matching M more popular than a matching N if the number of nodes preferring M over N is larger than the number of nodes preferring N over M. We call a matching popular, if there is no matching which is more popular. We characterize the instances with a popular matching, decide the existence of a popular matching, and compute a popular matching (if one exists) in time O(\(\sqrt{nm}\)) [AIKM05].
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We have a bipartite graph. The vertices on both sides rank the edges incident to them with ties allowed. A matching M is stable if there is no pair \((a, b) \in E{\backslash}M\) such that a prefers b over her mate in M and b prefers a over his mate in M or is indifferent between a and his mate. We show how to compute stable matchings in time O(nm) [KMMP04].
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In a random graph, edges are present with probability p independent of other edges. We show that for p ≥ c 0/n and c 0 a suitable constant, every non-maximal matching has a logarithmic length augmenting path. As a consequence the average running time of matching algorithms on random graphs is O(m log n) [BMSH05].
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References
Abraham, D., Irving, R., Kavitha, T., Mehlhorn, K.: Popular Matchings. In: SODA, pp. 424–432 (2005) ( http://www.mpi-sb.mpg.de/~mehlhorn/ftp/PopularMatchings.ps )
Bast, H., Mehlhorn, K., Schäfer, G., Tamaki, H.: Matching Algorithms are Fast in Sparse Random Graphs. Theory of Computing Systems 31(1), 3–14 (2005) (preliminary version). In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 81–92. Springer, Heidelberg (2004)
Irving, R., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-Maximal Matchings (A preliminary version appeared in SODA 2004, pp. 68–75). ACM Transactions on Algorithms 2(4), 1–9 (2006), http://www.mpi-sb.mpg.de/~mehlhorn/ftp/RankMaximalMatchings
Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Strongly Stable Matchings in Time O(nm) and Extensions to the Hospitals-Residents Problem (full version to appear in TALG). In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 222–233. Springer, Heidelberg (2004), http://www.mpi-sb.mpg.de/~mehlhorn/ftp/StableMatchings.ps
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Mehlhorn, K. (2007). Matchings in Graphs Variations of the Problem. In: Dress, A., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2007. Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73556-4_1
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DOI: https://doi.org/10.1007/978-3-540-73556-4_1
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