Smoothed performance guarantees for local search
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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 Classification68Q25 68Q87 68W25 68W40 90B35
We thank three anonymous referees for their valuable comments and suggestions that helped to improve the writing of the paper.
- 1.Angel, E.: A survey of approximation results for local search algorithms. In: Bampis, E., Jansen, K., Kenyon, C. (eds.) Efficient Approximation and Online Algorithms, Volume 3484 of LNCS, pp. 30–73. Springer, Heidelberg (2006)Google Scholar
- 7.Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1) article 4 (2007)Google Scholar
- 8.Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1295–13004 (2007)Google Scholar
- 15.Hoefer, M., Souza, A.: Tradeoffs and average-case equilibria in selfish routing. ACM Trans. Comput. Theory 2(1) article 2 (2010)Google Scholar
- 19.Michiels, W.P.A.J., Aarts, E.H.L., Korst, J.H.M.: Theoretical Aspects of Local Search. Springer, Heidelberg (2007)Google Scholar
- 22.Schäfer, G., Sivadasan, N.: Topology matters: smoothed competitiveness of metrical task systems. Theor. Comput. Sci. 341(1–3), 3–14 (2005)Google Scholar
- 26.Vöcking, B.: Selfish load balancing. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, chapter 20. Cambridge University Press, New York (2007)Google Scholar