On Budget-Constrained Flow Improvement

This paper investigates the complexity of budget-constrained flow improvement problems. We are given a directed graph with capacities on the edges which can be increased at linear costs up to some upper bounds. The problem is to increase the capacities within budget restrictions such that the flow from the source to the sink vertex is maximized. We show that the problem can be solved in polynomial time even if the improvement strategy is required to be integral. On the other hand, if the capacity of an edge must either be increased to the upper bound or left unchanged, then the problem turns NP-hard even on series-parallel graphs and strongly NP-hard on bipartite graphs. For the class series-parallel graphs we provide a fully polynomial approximation scheme for this problem