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Main Title: Complexity and Approximability of k-Splittable Flows
Author(s): Koch, Ronald
Spenke, Ines
Type: Research Paper
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 (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.
Subject(s): network flows
k-splittable flows
s, t-flow
Issue Date: 2005
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 90C27 Combinatorial optimization
90C60 Abstract computational complexity for mathematical programming problems
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2005, 29
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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