Semidefinite and Linear Programming Integrality Gaps for Scheduling Identical Machines

  • Adam Kurpisz
  • Monaldo Mastrolilli
  • Claire Mathieu
  • Tobias MömkeEmail author
  • Victor Verdugo
  • Andreas Wiese
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)


Sherali-Adams [25] and Lovász-Schrijver [21] developed systematic procedures to strengthen a relaxation known as lift-and-project methods. They have been proven to be a strong tool for developing approximation algorithms, matching the best relaxations known for problems like Max-Cut and Sparsest-Cut. In this work we provide lower bounds for these hierarchies when applied over the configuration LP for the problem of scheduling identical machines to minimize the makespan. First we show that the configuration LP has an integrality gap of at least \(1024\text {/}1023\) by providing a family of instances with 15 different job sizes. Then we show that for any integer n there is an instance with n jobs in this family such that after \(\varOmega (n)\) rounds of the Sherali-Adams (\(\text {SA}\)) or the Lovász-Schrijver (\(\text {LS}_+\)) hierarchy the integrality gap remains at least \(1024\text {/}1023\).


Approximation Algorithm Vertex Cover Polynomial Time Approximation Scheme Convex Relaxation Identical Machine 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Adam Kurpisz
    • 1
  • Monaldo Mastrolilli
    • 1
  • Claire Mathieu
    • 2
  • Tobias Mömke
    • 3
    Email author
  • Victor Verdugo
    • 2
    • 4
  • Andreas Wiese
    • 5
  1. 1.Dalle Molle Institute for Artificial Intelligence ResearchMannoSwitzerland
  2. 2.Department of Computer ScienceCNRS UMR 8548, École normale supérieureParisFrance
  3. 3.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  4. 4.Department of Industrial EngineeringUniversidad de ChileSantiagoChile
  5. 5.Max Planck Institute for InformaticsSaarbrückenGermany

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