Complexity and approximability of k-splittable flows

Let G=(V,E) be a graph with a source node s and a sink node t, |V|=n,|E|=m. For a given number k, the Maximum k-Splittable Flow problem (MkSF) is to find an s,t-flow of maximum value with a flow decomposition using at most k paths. In the multicommodity case this problem generalizes disjoint paths problems and unsplittable flow problems.We provide a comprehensive overview of the complexity and approximability landscape of MkSF on directed and undirected graphs. We consider constant values of k and k depending on graph parameters. For arbitrary constant values of k, we prove that the problem is strongly NP-hard on directed and undirected graphs already for k=2. This extends a known NP-hardness result for directed graphs that could not be applied to undirected graphs. Furthermore, we show that MkSF cannot be approximated with a performance ratio better than 5/6. This is the first constant bound given for arbitrary constant values of k. For non-constant values of k, the polynomial solvability was known before for all k⩾m, but open for smaller k. We prove that MkSF is NP-hard for all k fulfilling 2⩽k⩽m-n+1 (for n⩾3). For all other values of k the problem is shown to be polynomially solvable.