Dynamic matchings in left vertex weighted convex bipartite graphs - Zu, Quan; Zhang, Miaomiao; Yu, Bin (Journal of combinatorial optimization, 2016)
A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph G = ( X , Y , E ) in which the neighbors of each x ∈ X form an interval in Y where Y is linearly ordered, and each x ∈ 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.