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Approximation Algorithms for Scheduling with Reservations

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Abstract

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C max, which is the maximum completion time.

The general case of the problem is inapproximable unless \(\mathsf {P}=\mathsf {NP}\) ; hence, we study a suitable strongly \(\mathsf {NP}\) -hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemented by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless \(\mathsf {P}=\mathsf {NP}\) . The PTAS presented here is the first one for the problem under consideration; so far, not even for well-known special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a constant approximation ratio and discuss FPTASes for special cases as well as the complexity of the problem if m is part of the input.

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Author information

Correspondence to Florian Diedrich.

Additional information

An extended abstract of this work has been accepted at the 14th International Conference on High Performance Computing, HiPC 2007, Goa, India, December 18–21, 2007.

F. Diedrich’s research was supported in part by a grant “DAAD Doktorandenstipendium” of the German Academic Exchange Service and in part by EU research project AEOLUS, Algorithmic Principles for Building Efficient Overlay Computers, EU contract number 015964. Part of this work done while visiting the LIG, Grenoble University. Supported in part by DFG priority program 1126, “Algorithmics of Large and Complex Networks”; furthermore supported in part by a PPP funding “Scheduling in Communication Networks” D/05/06936 granted by the DAAD.

K. Jansen is supported in part by DFG priority program 1126, “Algorithmics of Large and Complex Networks” and in part by EU research project AEOLUS, Algorithmic Principles for Building Efficient Overlay Computers, EU contract number 015964.

Part of D. Trystram’s work was supported by the “CoreGRID” Network of Excellence.

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Diedrich, F., Jansen, K., Pascual, F. et al. Approximation Algorithms for Scheduling with Reservations. Algorithmica 58, 391–404 (2010). https://doi.org/10.1007/s00453-008-9271-2

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Keywords

  • Scheduling
  • Approximation algorithms
  • Reservations
  • Approximation scheme
  • PTAS
  • Complexity
  • Inapproximability