Matchings in Node-Weighted Convex Bipartite Graphs

Matchings in convex bipartite graphs correspond to the problem of scheduling unit–length tasks on a single disjunctive resource, given a release time and a deadline for every task. The unweighted case was studied by several authors since Glover first considered the problem in 1967 and until 1996, when Steiner and Yeomans found an algorithm whose running time is linear in the number of tasks. We address a weighted variant of this problem. Given a node–weighted convex bipartite graph G=(X, Y, E) (where Y is linearly ordered and the neighborhood of each node of X is an interval of Y), we show that it is possible to find an X–perfect matching of maximum (or minimum) weight in O(|E| + |Y| log |X|) time and O(|X| + |Y|) space. For the case in which only the nodes of Y are weighted and their weights are positive, the algorithm can be used to find a maximum–weight (not necessarily X–perfect) matching.