Dynamic matchings in left vertex weighted convex bipartite graphs

Abstract

A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph $G=(X,Y,E)$ in which the neighbors of each $x\in X$ form an interval in $Y$ where $Y$ is linearly ordered, and each $x\in X$ has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a dynamic LWCBG. The graph is subject to the updates of vertex and edge insertions and deletions. Our dynamic algorithms maintain the update operations in $O(\log ^2{|V|})$ amortized time per update, obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case $O(k)$ time, where $k$ is not greater than the cardinality of the maximum weight matching. That achieves the same time bound as the best known solution for the problem of the unweighted version.