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A Preemptive Algorithm for Maximizing Disjoint Paths on Trees

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Abstract

We consider the on-line version of the maximum vertex disjoint path problem when the underlying network is a tree. In this problem, a sequence of requests arrives in an on-line fashion, where every request is a path in the tree. The on-line algorithm may accept a request only if it does not share a vertex with a previously accepted request. The goal is to maximize the number of accepted requests. It is known that no on-line algorithm can have a competitive ratio better than Ω(log n) for this problem, even if the algorithm is randomized and the tree is simply a line. Obviously, it is desirable to beat the logarithmic lower bound. Adler and Azar (Proc. of the 10th ACM-SIAM Symposium on Discrete Algorithm, pp. 1–10, 1999) showed that if preemption is allowed (namely, previously accepted requests may be discarded, but once a request is discarded it can no longer be accepted), then there is a randomized on-line algorithm that achieves constant competitive ratio on the line. In the current work we present a randomized on-line algorithm with preemption that has constant competitive ratio on any tree. Our results carry over to the related problem of maximizing the number of accepted paths subject to a capacity constraint on vertices (in the disjoint path problem this capacity is 1). Moreover, if the available capacity is at least 4, randomization is not needed and our on-line algorithm becomes deterministic.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, 69978, Israel

    Yossi Azar

  2. Microsoft Research, Redmond, WA, 98052-6399, USA

    Yossi Azar

  3. Weizmann Institute, Rehovot, 76100, Israel

    Uriel Feige & Daniel Glasner

Authors
  1. Yossi Azar
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  2. Uriel Feige
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  3. Daniel Glasner
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Correspondence to Daniel Glasner.

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Research supported in part by the Israel Science Foundation.

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Azar, Y., Feige, U. & Glasner, D. A Preemptive Algorithm for Maximizing Disjoint Paths on Trees. Algorithmica 57, 517–537 (2010). https://doi.org/10.1007/s00453-009-9305-4

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  • DOI: https://doi.org/10.1007/s00453-009-9305-4

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