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A generalized approximation framework for fractional network flow and packing problems

Author

Listed:
  • Michael Holzhauser

    (University of Kaiserslautern)

  • Sven O. Krumke

    (University of Kaiserslautern)

Abstract

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $$\max \{c^T x : Ax \le b, x \in C \}$$ max { c T x : A x ≤ b , x ∈ C } , where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.

Suggested Citation

  • Michael Holzhauser & Sven O. Krumke, 2018. "A generalized approximation framework for fractional network flow and packing problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 19-50, February.
    • Handle: RePEc:spr:mathme:v:87:y:2018:i:1:d:10.1007_s00186-017-0604-2
    DOI: 10.1007/s00186-017-0604-2
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    References listed on IDEAS

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    1. Michael Holzhauser & Sven O. Krumke & Clemens Thielen, 2016. "Budget-constrained minimum cost flows," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1720-1745, May.
    2. Kevin D. Wayne, 2002. "A Polynomial Combinatorial Algorithm for Generalized Minimum Cost Flow," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 445-459, August.
    3. Serge A. Plotkin & David B. Shmoys & Éva Tardos, 1995. "Fast Approximation Algorithms for Fractional Packing and Covering Problems," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 257-301, May.
    4. Michael D. Grigoriadis & Leonid G. Khachiyan, 1996. "Coordination Complexity of Parallel Price-Directive Decomposition," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 321-340, May.
    5. Nimrod Megiddo, 1979. "Combinatorial Optimization with Rational Objective Functions," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 414-424, November.
    6. Refael Hassin, 1992. "Approximation Schemes for the Restricted Shortest Path Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 36-42, February.
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