Mathematical Programming

, Volume 154, Issue 1–2, pp 533–562 | Cite as

Scheduling and fixed-parameter tractability

  • Matthias Mnich
  • Andreas WieseEmail author
Full Length Paper Series B


Fixed-parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions. However, even though fixed-parameter algorithms are appealing for many reasons, no such algorithms are known for many fundamental scheduling problems. In this paper we present the first fixed-parameter algorithms for classical scheduling problems such as makespan minimization, scheduling with job-dependent cost functions—one important example being weighted flow time—and scheduling with rejection. To this end, we identify crucial parameters that determine the problems’ complexity. In particular, we manage to cope with the problem complexity stemming from numeric input values, such as job processing times, which is usually a core bottleneck in the design of fixed-parameter algorithms. We complement our algorithms with \(\mathsf {W[1]}\)-hardness results showing that for smaller sets of parameters the respective problems do not allow fixed-parameter algorithms. In particular, our positive and negative results for scheduling with rejection explore a research direction proposed by Dániel Marx.


Scheduling Fixed-parameter tractability Integer linear programming 

Mathematics Subject Classification

68W05 90B35 



We thank the reviewers of IPCO 2014 and Mathematical Programming for their remarks on improving the presentation of the results. We further thank an anonymous reviewer of an earlier version for suggestions how to improve the algorithms in Sect. 4 and to prove Theorem 9, as well as Akiyoshi Shioura for helpful discussions about convexity. Finally, we would like to thank the anonymous reviewers for pointing out the IP formulation in Sect. 2.2 which is simpler than our original formulation.


  1. 1.
    Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., Sviridenko, M.: Approximation schemes for minimizing average weighted completion time with release dates. In: Proc. FOCS, pp. 32–43 (1999)Google Scholar
  2. 2.
    Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bansal, N., Dhamdhere, K.: Minimizing weighted flow time. ACM Trans. Algorithms 3(4), 39 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bansal, N., Pruhs, K.: The geometry of scheduling. In: Proc. FOCS, pp. 407–414 (2010)Google Scholar
  6. 6.
    Bansal, N., Pruhs, K.: Weighted geometric set multi-cover via quasi-uniform sampling. In: Algorithms-ESA, pp. 145–156. Springer, Berlin (2012)Google Scholar
  7. 7.
    Bessy, S., Giroudeau, R.: Some parametric complexity results on a coupled-task scheduling problem, personal communication (2013)Google Scholar
  8. 8.
    Bodlaender, H.L., Fellows, M.R.: \(W[2]\)-hardness of precedence constrained \(K\)-processor scheduling. Oper. Res. Lett. 18(2), 93–97 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Brauner, N., Crama, Y., Grigoriev, A., van de Klundert, J.: A framework for the complexity of high-multiplicity scheduling problems. J. Comb. Optim. 9(3), 313–323 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chekuri, C., Khanna, S.: Approximation schemes for preemptive weighted flow time. In: Proc. STOC, pp. 297–305 (2002)Google Scholar
  11. 11.
    Chekuri, C., Khanna, S., Zhu, A.: Algorithms for minimizing weighted flow time. In: Proc. STOC, pp. 84–93 (2001)Google Scholar
  12. 12.
    Chu, G., Gaspers, S., Narodytska, N., Schutt, A., Walsh, T.: On the complexity of global scheduling constraints under structural restrictions. In: Proc. IJCAI (2013)Google Scholar
  13. 13.
    Chudak, F.A., Hochbaum, D.S.: A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Oper. Res. Lett. 25(5), 199–204 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Coffman E.G. Jr., Garey, M.R., Johnson, D.S.: An application of bin-packing to multiprocessor scheduling. SIAM J. Comput. 7, 1–17 (1978)Google Scholar
  15. 15.
    Ebenlendr, T., Krčál, M., Sgall, J.: Graph balancing: a special case of scheduling unrelated parallel machines. In: Proc. SODA, pp. 483–490 (2008)Google Scholar
  16. 16.
    Engels, D.W., Karger, D.R., Kolliopoulos, S.G., Sengupta, S., Uma, R.N., Wein, J.: Techniques for scheduling with rejection. J. Algorithms 49(1), 175–191 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fellows, M.R., Gaspers, S., Rosamond, F.A.: Parameterizing by the number of numbers. Theory Comput. Syst. 50(4), 675–693 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fellows, M.R., Koblitz, N.: Fixed-parameter complexity and cryptography. In: San Juan, P.R. (ed.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes Comput. Sci., vol. 673, pp. 121–131 (1993)Google Scholar
  19. 19.
    Fellows, M.R., McCartin, C.: On the parametric complexity of schedules to minimize tardy tasks. Theor. Comput. Sci. 298(2), 317–324 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Friesen, D.K.: Tighter bounds for the multifit processor scheduling algorithm. SIAM J. Comput. 13, 170–181 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Co., San Francisco (1979)zbMATHGoogle Scholar
  22. 22.
    Goemans, M.X., Rothvoß, T.: Polynomiality for bin packing with a constant number of item types. In: Proc. SODA, pp. 830–839 (2014)Google Scholar
  23. 23.
    Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 263–269 (1969)Google Scholar
  24. 24.
    Heinz, S.: Complexity of integer quasiconvex polynomial optimization. J. Complex. 21(4), 543–556 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34, 144–162 (1987)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hoogeveen, H., Skutella, M., Woeginger, G.J.: Preemptive scheduling with rejection. Math. Program. 94(2–3, Ser. B), 361–374 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79(1), 39–49 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Köppe, M.: On the complexity of nonlinear mixed-integer optimization. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 533–557 (2012)Google Scholar
  29. 29.
    Langston, M.A.: Processor scheduling with improved heuristic algorithms. Ph.D. thesis, Texas A&M University (1981)Google Scholar
  30. 30.
    Lenstra, J., Kan, A.R., Brucker, P.: Complexity of machine scheduling problems. In: Studies in Integer Programming, Ann. Discrete Math., vol. 1, pp. 343–362 (1977)Google Scholar
  31. 31.
    Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1–3), 259–271 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Marx, D.: Fixed-parameter tractable scheduling problems. In: Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091), vol. 1, p. 86 (2011)Google Scholar
  33. 33.
    Marx, D., Schlotter, I.: Stable assignment with couples: parameterized complexity and local search. Discret. Optim. 8(1), 25–40 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. In: Proc. IPCO 2014, Lecture Notes Comput. Sci., vol. 8494, pp. 381–392 (2014)Google Scholar
  35. 35.
    Sahni, S.: Algorithms for scheduling independent tasks. J. ACM 23, 116–127 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program. 62(1–3), 461–474 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Smith, W.E.: Various optimizers for single-stage production. Nav. Res. Logist. Q. 3, 59–66 (1956)CrossRefGoogle Scholar
  38. 38.
    Svensson, O.: Santa claus schedules jobs on unrelated machines. SIAM J. Comput. 41(5), 1318–1341 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Sviridenko, M., Wiese, A.: Approximating the configuration-LP for minimizing weighted sum of completion times on unrelated machines. In: Proc. IPCO 2013, Lecture Notes Comput. Sci., vol. 7801, pp. 387–398 (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Cluster of Excellence MMCISaarbrückenGermany
  2. 2.Max-Planck-Institute for Computer ScienceSaarbrückenGermany

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