Minimizing worst-case and average-case makespan over scenarios


We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all scenarios in an explicitly given set. Each scenario is a subset of jobs that must be executed in that scenario. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and lower bounds on their approximability. We also consider some (easier) special cases. Combinatorial optimization problems under scenarios in general, and scheduling problems under scenarios in particular, have seen only limited research attention so far. With this paper, we make a step in this interesting research direction.

This is a preview of subscription content, log in to check access.


  1. 1.

    An alternative way of viewing the scenarios would be that every scenario specifies a |J|-tuple of processing times, where the processing time of job j in any scenario \(S_i\) equals either \(p_j\) or 0.


  1. Ausiello, G., Crescenzi, P., Kann, V., Gambosi, G., Marchetti-Spaccamela, A., & Protasi, M. (1999). Complexity and approximation: Combinatorial optimization problems and their approximability properties. Berlin: Springer.

    Google Scholar 

  2. Austrin, P., Håstad, J., & Guruswami, V. (2014). \((2+\epsilon )\)-SAT is NP-hard. In: Proceedings of 55th Annual IEEE Symposium on Foundations of Computer Science (pp. 1–10).

  3. Ben-Tal, A., & Nemirovski, A. (2002). Robust optimization—methodology and applications. Mathematical Programming, 92(3), 453–480.

    Article  Google Scholar 

  4. Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. New York: Springer Science & Business Media.

    Google Scholar 

  5. Chekuri, C., & Khanna, S. (2004). On multidimensional packing problems. SIAM Journal on Computing, 33(4), 837–851.

    Article  Google Scholar 

  6. Chen, L., Megow, N., Rischke, R., & Stougie, L. (2015). Stochastic and robust scheduling in the cloud. In: Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (pp. 175–186).

  7. Epstein, L., Levin, A., Marchetti-Spaccamela, A., Megow, N., Mestre, J., Skutella, M., & Stougie, L. (2012). Universal sequencing on an unreliable machine. SIAM Journal on Computing, 41(3), 565–586.

  8. Garey, M. R., & Johnson, D. S. (1990). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman & Co.

    Google Scholar 

  9. Goemans, M. X., & Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6), 1115–1145.

    Article  Google Scholar 

  10. Gupta, A., Pál, M., Ravi, R., & Sinha, A. (2011). Sampling and cost-sharing: Approximation algorithms for stochastic optimization problems. SIAM Journal on Computing, 40(5), 1361–1401.

    Article  Google Scholar 

  11. Håstad, J. (2001). Some optimal inapproximability results. Journal of the ACM, 48(4), 798–859.

    Article  Google Scholar 

  12. Jaillet, P. (1985). Probabilistic traveling salesman problems. Technical Report 185, Operations Research Center, MIT.

  13. Jaillet, P. (1988). A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research, 36, 929–936.

    Article  Google Scholar 

  14. Karloff, H. J., & Zwick, U. (1997). A 7/8-approximation algorithm for MAX 3SAT? In: Proceedings of 38th Annual IEEE Symposium on Foundations of Computer Science (pp. 406–415).

  15. Kasperski, A., Kurpisz, A., & Zieliński, P. (2012). Parallel machine scheduling under uncertainty. In: Proceedings of 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU) (pp. 74–83).

  16. Kasperski, A., Kurpisz, A., & Zieliński, P. (2013). Approximating the min-max (regret) selecting items problem. Information Processing Letters, 113, 23–29.

    Article  Google Scholar 

  17. Kasperski, A., & Zieliński, P. (2016). Single machine scheduling problems with uncertain parameters and the OWA criterion. Journal of Scheduling, 19, 177–190.

    Article  Google Scholar 

  18. Khot, S. (2002). On the power of unique 2-prover 1-round games. In: Proceedings of 34th ACM Symposium on Theory of Computing (pp. 767–775).

  19. Khot, S., Kindler, G., Mossel, E., & O’Donnell, R. (2007). Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing, 37(1), 319–357.

    Article  Google Scholar 

  20. Kleywegt, A., Shapiro, A., & de Mello, T. H. (2002). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2), 479–502.

    Article  Google Scholar 

  21. Lin, X., Janak, S., & Floudas, C. (2004). A new robust optimization approach for scheduling under uncertainty: I. bounded uncertainty. Computers and Chemical Engineering, 28, 1069–1085.

  22. Oriolo, G., Sanità, L., & Zenklusen, R. (2013). Network design with a discrete set of traffic matrices. Operations Research Letters, 41(4), 390–396.

    Article  Google Scholar 

  23. Pinedo, M. (2012). Theory, scheduling algorithms and systems. Berlin: Springer.

    Google Scholar 

  24. Wang, Z., & Chan, F. (2015). A robust replenishment and production control policy for a single-stage production/inventory system with inventory inaccuracy. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 45(2), 326–337.

    Article  Google Scholar 

  25. Zhang, J., Ye, Y., & Han, Q. (2004). Improved approximations for max set splitting and max NAE SAT. Discrete Applied Mathematics, 142(1–3), 133–149.

    Article  Google Scholar 

  26. Zwick, U. (1999). Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of 31st ACM Symposium on Theory of Computing (pp. 679–687).

Download references

Author information



Corresponding author

Correspondence to Suzanne van der Ster.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Feuerstein, E., Marchetti-Spaccamela, A., Schalekamp, F. et al. Minimizing worst-case and average-case makespan over scenarios. J Sched 20, 545–555 (2017).

Download citation


  • Job scheduling
  • Approximation algorithm
  • Makespan
  • Scenarios