  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.
