Matchings in graphs variations of the problem

Abstract

Many real-life optimization problems are naturally formulated as questions about matchings in (bipartite) graphs. We have a bipartite graph. The edge set is partitioned into classes E1, E2, ..., Er. For a matching M, let si be the number of edges in M∩ E i. A rank-maximal matching maximizes the vector (s1, s2,..., sr). We show how to compute a rank-maximal matching in time O(r√nm) [IKM+06]. - 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(√nm) [AIKM05]. - 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) ∈ E \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]. - In a random graph, edges are present with probability p independent of other edges. We show that for p ≥ c0/n and c0 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]. © Springer-Verlag Berlin Heidelberg 2007.