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 $i\in{1,2}$ can be split into a bounded number $k_i$ 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 $\alpha>1/2$. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even $k_i$ 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 $(k_1,k_2)$-splittable flow without chunk size restrictions for fixed demand ratios. 
Elke Eisenschmidt and Utz-Uwe Haus
cs.DS cs.DM