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Approximating a class of combinatorial problems with rational objective function

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Abstract

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a 0 + a 1 x 1 + . . . + a n x n subject to certain constraints to solve the problem of minimizing a rational function of the form (a 0 + a 1 x 1 + . . . + a n x n )/(b 0 + b 1 x 1 + . . . + b n x n ) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo’s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo’s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.

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Authors and Affiliations

  1. Department of Industrial Engineering, Universidad de Chile, Santiago, RM, Chile

    José R. Correa

  2. Department of Computer Science, Universidade de São Paulo, São Paulo, Brazil

    Cristina G. Fernandes & Yoshiko Wakabayashi

Authors
  1. José R. Correa
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  2. Cristina G. Fernandes
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  3. Yoshiko Wakabayashi
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Correspondence to José R. Correa.

Additional information

A preliminary version of this paper appeared in the Proceedings of SWAT 2006. Research partially supported by CNPq Prosul Proc. 490333/04-4, Proc. 307011/03–8, and 305702/07–6 (Brazil); ProNEx - FAPESP/CNPq Proc. 2003/09925-5 (Brazil); and CONICYT (Chile), grants Anillo en Redes ACT08 and Fondecyt 1060035.

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Correa, J.R., Fernandes, C.G. & Wakabayashi, Y. Approximating a class of combinatorial problems with rational objective function. Math. Program. 124, 255–269 (2010). https://doi.org/10.1007/s10107-010-0364-8

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  • DOI: https://doi.org/10.1007/s10107-010-0364-8

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