Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph G = (U, V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u ∈ U , the neighbors of u are consecutive in the ordering of V . An induced matching H of G is a matching such that no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n + m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u ∈ U the first and last neighbor in the ordering of V . Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n + m) time. All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of O(n).