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Competitive Online Multicommodity Routing

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Abstract

We study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that depends on the previously routed flow. We present a competitive analysis of this algorithm showing that for affine price functions this algorithm is  \(\frac{4K^{2}}{(1+K)^{2}}\) -competitive, where K is the number of commodities. For networks with two nodes and parallel arcs, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive.

We then investigate a variant in which the demands have to be routed unsplittably. In this case, it is NP-hard to compute the offline optimum. The variant of the greedy algorithm that produces unsplittable flows is \((3+2\sqrt{2})\) -competitive, and we prove a lower bound of 2 for the competitive ratio of any deterministic online algorithm.

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Authors and Affiliations

  1. Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Tobias Harks

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

    Stefan Heinz

  3. Institute for Mathematical Optimization, Technische Universität Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Marc E. Pfetsch

Authors
  1. Tobias Harks
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  2. Stefan Heinz
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  3. Marc E. Pfetsch
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Correspondence to Stefan Heinz.

Additional information

A preliminary version of this paper appeared in [20].

S. Heinz has been supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.

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Harks, T., Heinz, S. & Pfetsch, M.E. Competitive Online Multicommodity Routing. Theory Comput Syst 45, 533–554 (2009). https://doi.org/10.1007/s00224-009-9187-5

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  • DOI: https://doi.org/10.1007/s00224-009-9187-5

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