Complexity and approximability of k-splittable flows

Abstract

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 (M kSF) 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 M kSF 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 M kSF 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 M kSF 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.