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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${i \in \{1, 2\}}$$</EquationSource> </InlineEquation> can be split into a bounded number k <Subscript> i </Subscript> 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 k <Subscript> i </Subscript> 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 <Subscript>1</Subscript>, k <Subscript>2</Subscript>)-splittable flow without chunk size restrictions for fixed demand ratios. Copyright Springer-Verlag 2013