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 this value. 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.