Approximation algorithms and relaxations for a service provision problem on a telecommunication network

Dye, Shane; Stougie, Leen; Tomasgård, Asgeir

doi:10.1016/s0166-218x(02)00234-2

Cited by 10 publications

(7 citation statements)

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“…If all demands for services are known in advance, the problem is NP-hard in the ordinary sense and a fully polynomial-time approximation scheme exists [16].…”

Section: Scenario-based Modelmentioning

confidence: 99%

Approximation in two-stage stochastic integer programming

Romeijnders

Stougie

Vlerk³

2014

Surveys in Operations Research and Management Science

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h i g h l i g h t s• We survey complexity results for stochastic (integer) programming problems.• We survey convex approximations of stochastic integer programming objectives.• We survey approximation guarantees for stochastic integer programming problems. a b s t r a c t Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solution value. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. a r t i c l e i n f oHowever, over the last thirty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis.Recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most deterministic combinatorial optimization problems. Polynomial-time approximation algorithms and their performance guarantees for stochastic linear and integer programming problems have received increasing research attention only very recently.Approximation in the traditional stochastic programming sense will not be discussed in this chapter. The reader interested in this issue is referred to surveys on stochastic programming, like the Handbook on Stochastic Programming by Ruszczyński and Shapiro (2003) or the textbooks by Birge and Louveaux (1997), Kall and Wallace (1994), Prékopa (1995), and Shapiro et al. (2009. We concentrate on the studies of approximation algorithms in relation to computational complexity theory.With this survey we intend to give a flavor of the type of results existing in the literature on approximation algorithms in two-stage stochastic integer programming rather than a complete overview of the literature on the subject. We do so by exhibiting a representative selection of results, which we present in full detail. While presenting them we do not refer to the literature; these references, together with pointers to other relevant work in this field of research, are given in an extensive notes section at the end of the survey.

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“…If all demands for services are known in advance, the problem is NP-hard in the ordinary sense and a fully polynomial-time approximation scheme exists [16].…”

Section: Scenario-based Modelmentioning

confidence: 99%

Approximation in two-stage stochastic integer programming

Romeijnders

Stougie

Vlerk³

2014

Surveys in Operations Research and Management Science

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“…This suggests also the existence of a fully polynomial approximation scheme for the problem, a nice subject for future investigations. This subclass of problems is still NP-hard since the problem with only one scenario (the deterministic problem) is NP-hard [2]. We argued in the introduction that this problem is not only of academic interest: in the telecommunication problem the number of scenarios to describe the random peak event is very small relative to the total number of services.…”

Section: Computational Complexitymentioning

confidence: 99%

“…The decision problem with multiple resources that asks whether there is a solution for which all demand is met, is strongly NP-complete [3]. The deterministic optimization problem with multiple resources is strongly NP-hard even when the number of resources is fixed, [2]. The deterministic optimization problem with a single resource is NP-hard in the ordinary sense [2].…”

Section: Introductionmentioning

confidence: 99%

“…The deterministic optimization problem with multiple resources is strongly NP-hard even when the number of resources is fixed, [2]. The deterministic optimization problem with a single resource is NP-hard in the ordinary sense [2]. The same paper presents a fully-polynomial time approximation scheme for this problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the deterministic case the knapsack LP solution can be found in O(n) time by a median finding algorithm using the price per unit criterion [2]. Here a similar approach is not known.…”

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The stochastic single resource service‐provision problem

Dye

Stougie

Tomasgård

2003

Naval Research Logistics

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The service‐provision problem described in this paper comes from an application of distributed processing in telecommunications networks. The objective is to maximize a service provider's profit from offering computational‐based services to customers. The service provider has limited capacity and must choose which of a set of software applications he would like to offer. This can be done dynamically, taking into consideration that demand for the different services is uncertain. The problem is examined in the framework of stochastic integer programming. Approximations and complexity are examined for the case when demand is described by a discrete probability distribution. For the deterministic counterpart, a fully polynomial approximation scheme is known S. Dye, L. Stougie, and A. Tomasgard, Approximation algorithms and relaxations for a service provision problem on a telecommunication network, Working Paper #2‐98, Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim, Norway, 1998, Discrete Appl Math, to appear. We show that introduction of stochasticity makes the problem strongly NP‐hard, implying that the existence of such a scheme for the stochastic problem is highly unlikely. For the general case a heuristic with a worst‐case performance ratio that increases in the number of scenarios is presented. Restricting the class of problem instances in a way that many reasonable practical problem instances satisfy allows for the derivation of a heuristic with a constant worst‐case performance ratio. Worst‐case performance analysis of approximation algorithms is classical in the field of combinatorial optimization, but in stochastic programming the authors are not aware of any previous results in this direction. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003

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Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

Ravi

Sinha

2004

Integer Programming and Combinatorial Optimization

107

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We study two-stage, finite-scenario stochastic versions of several combinatorial optimization problems, and provide nearly tight approximation algorithms for them. Our problems range from the graph-theoretic (shortest path, vertex cover, facility location) to set-theoretic (set cover, bin packing), and contain representatives with different approximation ratios.The approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart. Furthermore, we show that common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be adapted to obtain these results.

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Approximation algorithms and relaxations for a service provision problem on a telecommunication network

Cited by 10 publications

References 10 publications

Approximation in two-stage stochastic integer programming

Approximation in two-stage stochastic integer programming

The stochastic single resource service‐provision problem

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

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