Complexity and approximability of the maximum flow problem with minimum quantities

We consider the maximum flow problem with minimum quantities (MFPMQ), which is a variant of the maximum flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound (a minimum quantity), which may depend on the arc. This problem has recently been shown to be weakly NP -complete even on series-parallel graphs. In this article, we provide further complexity and approximability results for MFPMQ and several special cases. We first show that it is strongly NP -hard to approximate MFPMQ on general graphs (and even bipartite graphs) within any positive factor. On series-parallel graphs, however, we present a pseudo-polynomial time dynamic programming algorithm for the problem. We then study the case that the minimum quantity is the same for each arc in the network and show that, under this restriction, the problem is still weakly NP -complete on general graphs, but can be solved in strongly polynomial time on series-parallel graphs. On general graphs, we present (2-1/λ)-approximation algorithm for this case, where λ denotes the common minimum quantity of all arcs.