A Polynomial Time Approximation Algorithm for the Two-Commodity Splittable Flow Problem

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity i ∈ {1,2} can be split into a bounded number ki of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even ki and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k1,k2)-splittable flow without chunk size restrictions for fixed demand ratios.