Fast primal-dual update against local weight update in linear assignment problem and its application

Abstract

We consider a dynamic situation in the weighted bipartite matching problem: edge weights in the input graph are repeatedly updated and we are asked to maintain an optimal matching at any moment. A trivial approach is to compute an optimal matching from scratch each time an update occurs. In this paper, we show that if each update occurs locally around a single vertex, then a single execution of Dijkstra's algorithm is sufficient to preserve optimality with the aid of a dual solution. As an application of our result, we provide a faster implementation of the envy-cycle procedure for finding an envy-free allocation of indivisible items. Our algorithm runs in O(mn2) time, while the known bound of the original one is O(mn3), where n and m denote the numbers of agents and items, respectively.