Strong LP Formulations for Scheduling Splittable Jobs on Unrelated Machines

  • José R. Correa
  • Alberto Marchetti-Spaccamela
  • Jannik Matuschke
  • Leen Stougie
  • Ola Svensson
  • Víctor Verdugo
  • José Verschae
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

We study a natural generalization of the problem of minimizing makespan on unrelated machines in which jobs may be split into parts. The different parts of a job can be (simultaneously) processed on different machines, but each part requires a setup time before it can be processed. First we show that a natural adaptation of the seminal approximation algorithm for unrelated machine scheduling [11] yields a 3-approximation algorithm, equal to the integrality gap of the corresponding LP relaxation. Through a stronger LP relaxation, obtained by applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ ≈ 1.618 is the golden ratio. This ratio decreases to 2 in the restricted assignment setting, matching the result for the classic version. Interestingly, we show that our problem cannot be approximated within a factor better than \(\frac{e}{e-1}\approx 1.582\) (unless \(\mathcal{P}=\mathcal{NP}\)). This provides some evidence that it is harder than the classic version, which is only known to be inapproximable within a factor 1.5 − ε. Since our 1 + φ bound remains tight when considering the seemingly stronger machine configuration LP, we propose a new job based configuration LP that has an infinite number of variables, one for each possible way a job may be split and processed on the machines. Using convex duality we show that this infinite LP has a finite representation and can be solved in polynomial time to any accuracy, rendering it a promising relaxation for obtaining better algorithms.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • José R. Correa
    • 1
  • Alberto Marchetti-Spaccamela
    • 2
  • Jannik Matuschke
    • 1
  • Leen Stougie
    • 3
  • Ola Svensson
    • 4
  • Víctor Verdugo
    • 1
  • José Verschae
    • 1
  1. 1.Departamento de Ingeniería IndustrialUniversidad de ChileChile
  2. 2.Department of Computer and System SciencesSapienza University of RomeItaly
  3. 3.Department of Econometrics and Operations ResearchVU Amsterdam & CWIThe Netherlands
  4. 4.School of Computer and Communication SciencesEPFLSwitzerland

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