Mathematical Programming

, Volume 146, Issue 1–2, pp 185–218 | Cite as

Smoothed performance guarantees for local search

  • Tobias Brunsch
  • Heiko Röglin
  • Cyriel Rutten
  • Tjark VredeveldEmail author
Full Length Paper Series A


We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are \(\Theta (\phi )\) and \(\Theta (\log \phi )\), respectively, where \(1/\phi \) is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is \(\Theta (\log \phi )\) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of  \(\Theta (\log \phi )\).

Mathematics Subject Classification

68Q25 68Q87 68W25 68W40 90B35 



We thank three anonymous referees for their valuable comments and suggestions that helped to improve the writing of the paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  • Tobias Brunsch
    • 1
  • Heiko Röglin
    • 1
  • Cyriel Rutten
    • 2
  • Tjark Vredeveld
    • 2
    Email author
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany
  2. 2.Department of Quantitative EconomicsMaastricht UniversityMaastrichtThe Netherlands

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