Scheduling meets n-fold integer programming


Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In this paper, we continue this study and show that several additional cases of fundamental scheduling problems are fixed-parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where \(p_{\max }\) is the maximum processing time of a job and \(w_{\max }\) is the maximum weight of a job:

  • Makespan minimization on uniformly related machines (\(Q||C_{\max }\)) parameterized by \(p_{\max }\),

  • Makespan minimization on unrelated machines (\(R||C_{\max }\)) parameterized by \(p_{\max }\) and the number of kinds of machines (defined later),

  • Sum of weighted completion times minimization on unrelated machines (\(R||\sum w_jC_j\)) parameterized by \(p_{\max }+w_{\max }\) and the number of kinds of machines,

  • The same problem, \(R||\sum w_jC_j\), parameterized by the number of distinct job times and the number of machines.

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  1. 1.

    Theorem 4.1 does not explicitly mention the step length \(\gamma \), but it is implicit in the proof that even this stronger statement holds. Also, Theorem 4.1 does not give the complexity bound, which is contained in Theorem 6.2.)

  2. 2.

    For example, if, for any \(\alpha \in \mathbb {R}\), there is a separation oracle for the level set \(\{\mathbf{x}\mid f(\mathbf{x}) \le \alpha \}\).


  1. Allahverdi, A. (2015). The third comprehensive survey on scheduling problems with setup times/costs. European Journal of Operational Research, 246(2), 345–378.

    Article  Google Scholar 

  2. Asahiro, Y., Jansson, J., Miyano, E., Ono, H., & Zenmyo, K. (2011). Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. Journal of Combinatorial Optimization, 22(1), 78–96.

    Article  Google Scholar 

  3. Blekherman, G., Parrilo, P. A., & Thomas, R. R. (2012). Semidefinite optimization and convex algebraic geometry. Philadelphia: SIAM.

    Google Scholar 

  4. Bodlaender, H. L., & Fellows, M. R. (1995). W[2]-hardness of precedence constrained k-processor scheduling. Operations Research Letter, 18(2), 93–97.

    Article  Google Scholar 

  5. Bruno, J., Coffman, E. G, Jr., & Sethi, R. (1974). Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7), 382–387.

    Article  Google Scholar 

  6. Chen, L., Marx, D., Ye, D., & Zhang, G. (2017). Parameterized and approximation results for scheduling with a low rank processing time matrix. In LIPIcs-Leibniz international proceedings in informatics (Vol. 66). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

  7. Demaine, E.D., Hajiaghayi, M., Marx, D. (eds.) (2009). Parameterized complexity and approximation algorithms. Dagstuhl seminar proceedings (Vol. 09511), 13.12.2009–17.12.2009, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany

  8. Downey, R. G., & Fellows, M. R. (2013). Fundamentals of parameterized complexity. Texts in computer science. Berlin: Springer.

    Google Scholar 

  9. Fellows, M. R., & McCartin, C. (2003). On the parametric complexity of schedules to minimize tardy tasks. Theoretical Computer Science, 2(298), 317–324.

    Article  Google Scholar 

  10. Garey, M. R., & Johnson, D. S. (1979) Computers and intractability: A guide to the theory of np-completeness

  11. Goemans, M., & Williamson, D. P. (2000). Two-dimensional Gantt charts and a scheduling algorithm of Lawler. SIAM Journal on Discrete Mathematics, 13(3), 281–294.,

    Article  Google Scholar 

  12. Halldórsson, M.M., & Karlsson, R.K. (2006). Strip graphs: Recognition and scheduling. In WG 2006 (pp. 137–146).

    Google Scholar 

  13. Hemmecke, R., Köppe, M., & Weismantel, R. (2014). Graver basis and proximity techniques for block-structured separable convex integer minimization problems. Mathematical Programming, 145(1–2), 1–18.

    Article  Google Scholar 

  14. Hemmecke, R., Onn, S., & Romanchuk, L. (2013). n-fold integer programming in cubic time. Mathematical Programming, 137(1–2), 325–341.

    Article  Google Scholar 

  15. Hermelin, D., Kubitza, J., Shabtay, D., Talmon, N., & Woeginger, G. J. (2015). Scheduling two competing agents when one agent has significantly fewer jobs. IPEC, 2015, 55–65.

    Article  Google Scholar 

  16. Hildebrand, R., & Köppe, M. (2013). A new Lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity \(2^{O(n{\rm log}n)}\). Discrete Optimization, 10(1), 69–84.

    Article  Google Scholar 

  17. Hochbaum, D. S., & Shanthikumar, J. G. (1990). Convex separable optimization is not much harder than linear optimization. Journal of the ACM, 37(4), 843–862.

    Article  Google Scholar 

  18. Horn, W. A. (1973). Technical note—Minimizing average flow time with parallel machines. Operations Research, 21(3), 846–847.

    Article  Google Scholar 

  19. Jansen, K., Kratsch, S., Marx, D., & Schlotter, I. (2013). Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1), 39–49.

    Article  Google Scholar 

  20. Khachiyan, L., & Porkolab, L. (2000). Integer optimization on convex semialgebraic sets. Discrete & Computational Geometry, 23(2), 207–224.

    Article  Google Scholar 

  21. Knop, D., Kouteckỳ, M., & Mnich, M. (2017) Voting and bribing in single-exponential time. In 34th symposium on theoretical aspects of computer science.

  22. Kononov, A. V., Sevastyanov, S., & Sviridenko, M. (2012). A complete 4-parametric complexity classification of short shop scheduling problems. Journal of Scheduling, 15(4), 427–446.

    Article  Google Scholar 

  23. Lawler, E. L., Lenstra, J. K., Kan, A. H. R., & Shmoys, D. B. (1993). Sequencing and scheduling: Algorithms and complexity. Handbooks in operations research and management science, 4, 445–522.

    Article  Google Scholar 

  24. Lenstra, H. W, Jr. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 538–548.

    Article  Google Scholar 

  25. Loera, J. A. D., Hemmecke, R., & Köppe, M. (2013). Algebraic and geometric ideas in the theory of discrete optimization. MOS-SIAM series on optimization (Vol. 14). Philadelphia: SIAM.

    Google Scholar 

  26. Marx, D. (2011). Packing and scheduling algorithms for information and communication services (dagstuhl seminar 11091). Dagstuhl Reports, 1(2), 67–93.

    Article  Google Scholar 

  27. Mnich, M., & Wiese, A. (2014). Scheduling and fixed-parameter tractability. Mathematical Programming, 154(1), 533–562.

    Article  Google Scholar 

  28. Onn, S. (2010). Nonlinear discrete optimization. Zurich Lectures in Advanced Mathematics: European Mathematical Society.

  29. Onn, S., & Sarrabezolles, P. (2015). Huge unimodular \(n\)-fold programs. SIAM Journal on Discrete Mathematics, 29(4), 2277–2283.

    Article  Google Scholar 

  30. Potts, C. N., & Strusevich, V. A. (2009). Fifty years of scheduling: A survey of milestones. Journal of the Operational Research Society, 60(1), S41–S68.

    Article  Google Scholar 

  31. Sitters, R. (2005). Complexity of preemptive minsum scheduling on unrelated parallel machines. Journal of Algorithms, 57(1), 37–48.

    Article  Google Scholar 

  32. van Bevern, R., Bredereck, R., Bulteau, L., Komusiewicz, C., Talmon, N., & Woeginger, G.J. (2016). Precedence-constrained scheduling problems parameterized by partial order width. In Proceedings of the 9th international conference on discrete optimization and operations research, DOOR 2016, Vladivostok, Russia, September 19–23, 2016, pp. 105–120.

  33. van Bevern, R., Mnich, M., Niedermeier, R., & Weller, M. (2015a). Interval scheduling and colorful independent sets. Journal of Scheduling, 18(5), 449–469.

    Article  Google Scholar 

  34. van Bevern, R., Niedermeier, R., & Suchý, O. (2015b). A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: Few machines, small looseness, and small slack. CoRR arXiv:1508.01657.

  35. van Bevern, R., & Pyatkin, A. V. (2016). Completing partial schedules for open shop with unit processing times and routing. CSR, 2016, 73–87.

    Article  Google Scholar 

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We would like to thank René van Bevern for pointing us to much related work. We are grateful to Matthias Mnich for pointing out the exponential speed-up of our result compared to the previous work, and other useful comments. Finally, we are grateful to the anonymous reviewers for their comments which made the paper more readable.

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Correspondence to Dušan Knop.

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This research was partially supported by the Project 17-09142S of GA ČR, the CE-ITI grant Project P202/12/G061 of GA ČR, Project SVV-2017-260452 and Projects 1784214 and 338216 GA UK.

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Knop, D., Koutecký, M. Scheduling meets n-fold integer programming. J Sched 21, 493–503 (2018).

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  • Fixed parameterized tractability
  • Scheduling on parallel machines

Mathematics Subject Classification

  • 90B35
  • 90C10
  • 03D15