Smoothed performance guarantees for local search


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  \(\Theta (\log \phi )\).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 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)

  2. 2.

    Aspnes, J., Azar, Y., Fiat, A., Plotkin, S.A., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44(3), 486–504 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. Theor. Comput. Sci. 361, 200–209 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Schäfer, G., Vredeveld, T.: Average case and smoothed competitive analysis for the multi-level feedback algorithm. Math. Oper. Res. 31(3), 85–108 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. Syst. Sci. 69(3), 306–329 (2004)

    Article  MATH  Google Scholar 

  6. 6.

    Cho, Y., Sahni, S.: Bounds for list schedules on uniform processors. SIAM J. Comput. 9, 91–103 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1) article 4 (2007)

    Google Scholar 

  8. 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)

  9. 9.

    Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19, 312–320 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Garey, M.R., Johnson, D.S.: Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  11. 11.

    Glass, C.A., Kellerer, H.: Parallel machine scheduling with job assignment restrictions. Naval Res. Logist. 54(3), 250–257 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)

    Article  Google Scholar 

  13. 13.

    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Hochbaum, D.S., Shmoys, D.B.: A polynomial approximation scheme for machine scheduling on uniform processors: using the dual approximation approach. SIAM J. Comput. 17, 539–551 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Hoefer, M., Souza, A.: Tradeoffs and average-case equilibria in selfish routing. ACM Trans. Comput. Theory 2(1) article 2 (2010)

    Google Scholar 

  16. 16.

    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Leung, J.Y.T., Li, C.L.: Scheduling with processing set restrictions: a survey. Int. J. Prod. Econ. 116, 251–262 (2008)

    Article  Google Scholar 

  18. 18.

    Li, C.L.: Scheduling unit-length jobs with machine eligibility restrictions. Eur. J. Oper. Res. 174, 1325–1328 (2006)

    Article  MATH  Google Scholar 

  19. 19.

    Michiels, W.P.A.J., Aarts, E.H.L., Korst, J.H.M.: Theoretical Aspects of Local Search. Springer, Heidelberg (2007)

  20. 20.

    Ou, J., Leung, J.Y.-T., Li, C.L.: Scheduling parallel machines with inclusive set restrictions. Naval Res. Logist. 55(4), 328–338 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Rutten, C., Recalde, D., Schuurman, P., Vredeveld, T.: Performance guarantees of jump neighborhoods on restricted related parallel machines. Oper. Res. Lett. 40, 287–291 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Schäfer, G., Sivadasan, N.: Topology matters: smoothed competitiveness of metrical task systems. Theor. Comput. Sci. 341(1–3), 3–14 (2005)

    Google Scholar 

  23. 23.

    Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. Inf. J. Comput. 19(1), 52–63 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Spielman, D.A., Teng, S.H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Spielman, D.A., Teng, S.H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)

    Article  Google Scholar 

  26. 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)

Download references


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

Author information



Corresponding author

Correspondence to Tjark Vredeveld.

Additional information

A preliminary version of this paper appeared in the proceedings of ESA 2011.


Appendix A: Table of notation

In the table below, the notation used in this paper is summarized.

\(J\) Set of jobs \(1, \ldots , n\)
\(M\) Set of machines \(1, \ldots , m\)
\(p_j\) Processing requirement of job \(j\)
\(s_i\) Speed of machine \(i\)
\(\mathcal{M }_j\) Set of machines on which job \(j\) can be scheduled
\(s_{\max }\) Maximum speed of the machines
\(s_{\min }= 1\) Minimum speed of the machines;
     by scaling we assume w.l.o.g. it to be \(1\)
\({C_{\max }^{*}}\) Optimal makespan
\({C_{\max }}({\sigma })\) Makespan of schedule \({\sigma }\)
\({J_{i}}({\sigma })\) Set of jobs scheduling on machine \(i\) in schedule \({\sigma }\)
\({L_{i}}({\sigma })\) \( = \sum _{j \in {J_{i}}({\sigma })} p_j / s_i\)
     load of machine \(i\) in schedule \({\sigma }\)
\({J_{i,j}}({\sigma })\) \( = {J_{i}}({\sigma }) \cap \{ 1, \ldots , j \}\)
\(j_i^t\) \(= \min \left\{ j \,:\,\sum _{\ell \in {J_{i,j}}({\sigma })} p_{\ell } / s_i \ge t \cdot {C_{\max }^{*}}\right\} \)
\({J_{i,\ge t}}({\sigma })\) \(= {J_{i,j_i^t}}({\sigma })\)
\(c\) \( = \left\lfloor \frac{{C_{\max }}({\sigma })}{{C_{\max }^{*}}} \right\rfloor - 1\)
\(i_k\) \(= \max \left\{ i \in M \,:\,{L_{i^{\prime }}} \ge k \cdot {C_{\max }^{*}} \, \forall \, i^{\prime } \le i \right\} \),
       assuming \(s_1 \ge s_2 \ge \cdots \ge s_m\)
\(H_{k}\) \(= \{1, \ldots , i_k\}\)
\(R_k\) \(= H_k{\setminus }H_{k+1}\) for \(k=0,1,\ldots , c-1\)
\(R_c\) \(= H_c\)

Appendix B: Hoeffding’s bound

On several occasions in this paper we use Hoeffding’s bound [16] to bound tail probabilities. For completeness, we state the bound in the following theorem.

Theorem 31

Let \(X_1, \ldots , X_n\) be independent random variables. Define \(X := \sum _{j=1}^n X_j\) and \(\mu = {{\mathop {\mathbf{E}}}[X]}\). If each \(X_j \in [a_j,b_j]\) for some constants \(a_j\) and \(b_j\), \(j = 1,\ldots ,n\), then for any \(t > 0\)

$$\begin{aligned} {\mathop {\mathbf{Pr}}\limits _{}\left[ X \le {{\mathop {\mathbf{E}}}[X]} - t\right] }&\le \exp \left( \frac{-2t}{\sum _j (b_j - a_j)^2} \right) , \quad \text{ and, } \\ {\mathop {\mathbf{Pr}}\limits _{}\left[ X \ge {{\mathop {\mathbf{E}}}[X]} + t\right] }&\le \exp \left( \frac{-2t}{\sum _j (b_j - a_j)^2} \right) . \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brunsch, T., Röglin, H., Rutten, C. et al. Smoothed performance guarantees for local search. Math. Program. 146, 185–218 (2014).

Download citation

Mathematics Subject Classification

  • 68Q25
  • 68Q87
  • 68W25
  • 68W40
  • 90B35