Complexity of strict robust integer minimum cost flow problems: An overview and further results

Abstract

We consider a robust version of the integer minimum cost flow problem (IMCF) with uncertainty in the cost function, which is represented by a discrete scenario set. It is known that this problem is NP-hard in general. Besides general graphs, we analyze the problem complexities for acyclic and series-parallel graphs. For both classes we are able to present pseudo-polynomial algorithms when the flow value F and the size of the uncertainty set is fixed. On series-parallel graphs, this was even possible for unbounded F by using the recursive structure to develop a dynamic programming algorithm. For acyclic networks, we transfer the robust flow problem to a robust shortest path problem on a new graph. This graph can be deduced from the original network and the flow value F. Beside the theoretical studies, we also test practical ideas to improve the efficiency of the label setting algorithm, which we use to solve the robust shortest path problem. We improve the running time by using an upper bound and precomputed lower bounds in each vertex, similar to the A*-search.