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A generalized approximation framework for fractional network flow and packing problems

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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 \(\max \{c^T x : Ax \le b, x \in C \}\), 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.

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Notes

  1. We use the notation \(B_{l\cdot }\) for a matrix B to denote the l-th row of the matrix.

  2. We will see how we can “filter out” vectors \(x^{(l)}\) with negative costs in the following sections.

  3. The sign function \({{\mathrm{sgn}}}:\mathbb {R} \mapsto \{-1,0,1\}\) returns \(-1\), 0, or 1 depending on whether the argument is negative, zero, or positive, respectively.

  4. Actually, since we do not have direct access to the set S, we need to obtain such a vector via an oracle access. However, by calling the oracle once more with a very large value for \(\lambda \) or by returning some vector found before, we obtain a certificate in S, which we can return.

  5. M denotes the largest absolute value of each number given in the problem instance, assuming gain factor are given as ratios of integers.

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Authors and Affiliations

  1. Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 14, 67663, Kaiserslautern, Germany

    Michael Holzhauser & Sven O. Krumke

Authors
  1. Michael Holzhauser
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  2. Sven O. Krumke
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Correspondence to Michael Holzhauser.

Additional information

This work was partially supported by the German Federal Ministry of Education and Research within the project “SinOptiKom—Cross-sectoral Optimization of Transformation Processes in Municipal Infrastructures in Rural Areas”.

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Holzhauser, M., Krumke, S.O. A generalized approximation framework for fractional network flow and packing problems. Math Meth Oper Res 87, 19–50 (2018). https://doi.org/10.1007/s00186-017-0604-2

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  • DOI: https://doi.org/10.1007/s00186-017-0604-2

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