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Preprint 29-2005

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

Complexity and Approximability of k-Splittable Flows
primary: 90C27 Combinatorial optimization
secondary: 90C60 Abstract computational complexity for mathematical programming problems
network flows, k-splittable flows, complexity, approximability
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.
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