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Mathematical Programming

, Volume 154, Issue 1–2, pp 305–328 | Cite as

Strong LP formulations for scheduling splittable jobs on unrelated machines

  • José Correa
  • Alberto Marchetti-Spaccamela
  • Jannik Matuschke
  • Leen Stougie
  • Ola Svensson
  • Víctor Verdugo
  • José Verschae
Full Length Paper Series B

Abstract

A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to \(1+\phi \), where \(\phi \) is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than \(\frac{e}{e-1}\approx 1.582\, (\hbox {unless } \mathcal {P}=\mathcal {NP})\), which is larger than the inapproximability bound of 1.5 for the original problem.

Mathematics Subject Classification

Primary 90B35 68W25 Secondary 68Q25 90C10 

Notes

Acknowledgments

We would like to thank two anonymous referees for their insightful remarks, which helped improving the presentation of Sects. 4 and 6. This work was partially supported by Nucleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F, by EU-IRSES Grant EUSACOU, by the DFG Priority Programme “Algorithm Engineering” (SPP 1307), by ERC Starting Grant 335288-OptApprox, by FONDECYT Project 3130407, by the Berlin Mathematical School and by the Tinbergen Institute and ABRI-VU.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • José Correa
    • 1
  • Alberto Marchetti-Spaccamela
    • 2
  • Jannik Matuschke
    • 3
  • Leen Stougie
    • 4
  • Ola Svensson
    • 5
  • Víctor Verdugo
    • 1
  • José Verschae
    • 6
    • 7
  1. 1.Departamento de Ingeniería IndustrialUniversidad de ChileSantiagoChile
  2. 2.Department of Computer and System SciencesSapienza University of RomeRomeItaly
  3. 3.Institut für MathematikTU BerlinBerlinGermany
  4. 4.Department of Econometrics and Operations ResearchVU Amsterdam & CWIAmsterdamThe Netherlands
  5. 5.School of Computer and Communication SciencesEPFLLausanneSwitzerland
  6. 6.Departamento de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  7. 7.Departamento de Ingeniería Industrial y de SistemasPontificia Universidad Católica de ChileSantiagoChile

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