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# Approximation and complexity of k-splittable flows

## Koch, Ronald; Skutella, Martin; Spenke, Ines

## Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Given a graph with a source and a sink node, the maximum k-splittable flow (MkSF) problem is to find a flow of maximum value with a flow decomposition using at most k paths. In the more general context of multicommodity flows, this NP-hard problem has been introduced by Baier, Koehler, and Skutella. It generalizes disjoint paths problems and unsplittable flows. A particularly interesting aspect of this problem is that its solution requires both packing and routing strategies. The total flow has to be divided into k packets which then have to be routed along certain paths of the given graph. The main contributions of this paper are as follows: (i) We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal MkSF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we canguarantee that, for any fixed ǫ >0, the computed set of alternatives contains a packing used by a (1−ǫ)–approximate solution. The latter result is based on the observation that (1−ǫ)–approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. (ii) Based on (i), we prove that, for constant k, the MkSF problem can be solved in polynomial time on graphs of bounded tree width. If k is part of the input, this problem is still N P–hard and we present a polynomial time approximation scheme for it. (iii) Finally, we provide a comprehensive overview of the complexity and approximability landscape of MkSF for different values of k.