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Flows with unit path capacities and related packing and covering problems

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Abstract

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest.

Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an \(O(\sqrt{m})\) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.

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Authors and Affiliations

  1. Zuse Institute Berlin, Takustr. 7, 14195, Berlin, Germany

    Maren Martens

  2. TU Berlin, Institut für Mathematik, MA 5–2, Str. des 17. Juni 136, 10623, Berlin, Germany

    Martin Skutella

Authors
  1. Maren Martens
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  2. Martin Skutella
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Correspondence to Maren Martens.

Additional information

This work was partially supported by the DFG Research Center Matheon in Berlin, by the Graduate School of Production Engineering and Logistics, North Rhine-Westphalia, by the DFG Focus Program 1126 and the DFG grants SK 58/4-1 and SK 58/5-3, and by an NSERC Operating Grant.

Part of this work was done while M. Martens was at Universität Dortmund and at the Sauder School of Business, University of British Columbia.

Part of this work was done while M. Skutella was at Universität Dortmund.

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Martens, M., Skutella, M. Flows with unit path capacities and related packing and covering problems. J Comb Optim 18, 272–293 (2009). https://doi.org/10.1007/s10878-009-9225-x

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