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# Complexity and Approximability of k-Splittable Flows

## Koch, Ronald; Spenke, Ines

## Preprint-Reihe des Instituts fÃ¼r Mathematik, Technische UniversitÃ¤t Berlin

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 polynomially 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.