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Main Title: Maximum k-Splittable Flows
Author(s): Koch, Ronald
Skutella, Martin
Spenke, Ines
Type: Research Paper
Abstract: Given a graph with a source and a sink node, the NP-hard maximum k-splittable s,t-flow (MkSF) problem is to find a flow of maximum value from s to t with a flow decomposition using at most k paths. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k-splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values on these paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Our main contributions 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 can guarantee that, for any fixed epsilon>0, the computed set of alternatives contains a packing used by a (1-epsilon)-approximate solution. The latter result is based on the observation that (1-epsilon)-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 treewidth. If k is part of the input, this problem is still NP-hard and we present a polynomial time approximation scheme for it.
Subject(s): network flows
k-splittable flows
approximation algorithms
bounded treewidth
Issue Date: 2005
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 90C27 Combinatorial optimization
90C59 Approximation methods and heuristics
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2005, 30
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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