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Multicommodity Demand Flow in a Tree

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Automata, Languages and Programming (ICALP 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2719))

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Abstract

We consider requests for capacity in a given tree network T = (V,E) where each edge of the tree has some integer capacity u e. Each request consists of an integer demand d f and a profit w f which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provide a maximum profit. This generalizes well-known problems including the knapsack and b-matching problems.

When all demands are 1, we have the integer multicommodity flow problem. Garg, Vazirani, and Yannakakis [5] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. In this paper we establish for the first time that the natural linear programming relaxation has a constant factor gap, a factor of 4, for the case of arbitrary profits.

We then discuss the situation for arbitrary demands. When the maximum demand d max is at most the minimum edge capacity u min, we show that the integrality gap of the LP is at most 48. This result is obtained showing that the integrality gap for demand version of such a problem is at most 12 times that for the unit demand case. We use techniques of Kolliopoulos and Stein [8], [9] to obtain this. We also obtain, via this method, improved algorithms for the line and ring networks. Applications and connections to other combinatorial problems are discussed.

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References

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Author information

Authors and Affiliations

  1. Bell Labs, 600 Mountain Ave, Murray Hill, NJ, 07974

    Chandra Chekuri & F. Bruce Shepherd

  2. Computer Science Dept., Rutgers University, Piscataway, NJ, 08854-8019

    Marcelo Mydlarz

Authors
  1. Chandra Chekuri
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  2. Marcelo Mydlarz
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  3. F. Bruce Shepherd
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Jos C. M. Baeten

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA, 30332-0205, USA

    Jan Karel Lenstra

  3. Department of Information Technology, Uppsala University, P.O. Box 337, 75105, Uppsala, Sweden

    Joachim Parrow

  4. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Gerhard J. Woeginger

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© 2003 Springer-Verlag Berlin Heidelberg

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Chekuri, C., Mydlarz, M., Shepherd, F.B. (2003). Multicommodity Demand Flow in a Tree. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_34

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  • DOI: https://doi.org/10.1007/3-540-45061-0_34

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40493-4

  • Online ISBN: 978-3-540-45061-0

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