Journal of Scheduling

, Volume 15, Issue 4, pp 427–446 | Cite as

A Complete 4-parametric complexity classification of short shop scheduling problems

  • Alexander Kononov
  • Sergey Sevastyanov
  • Maxim Sviridenko


We present a comprehensive complexity analysis of classical shop scheduling problems subject to various combinations of constraints imposed on the processing times of operations, the maximum number of operations per job, the upper bound on schedule length, and the problem type (taking values “open shop,” “job shop,” “mixed shop”). It is shown that in the infinite class of such problems there exists a finite basis system that allows one to easily determine the complexity of any problem in the class. The basis system consists of ten problems, five of which are polynomially solvable, and the other five are NP-complete. (The complexity status of two basis problems was known before, while the status of the other eight is determined in this paper.) Thereby the dichotomy property of that parameterized class of problems is established. Since one of the parameters is the bound on schedule length (and the other two numerical parameters are tightly related to it), our research continues the research line on complexity analysis of short shop scheduling problems initiated for the open shop and job shop problems in the paper by Williamson et al. (Oper. Res. 45(2):288–294, 1997). We improve on some results of that paper.


Shop scheduling Makespan minimization Complexity classification 


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  1. Bansal, N., Mahdian, M., & Sviridenko, M. (2005). Minimizing makespan in no-wait job shops. Mathematics of Operations Research, 30(4), 817–831. CrossRefGoogle Scholar
  2. Baptiste, P., Carlier, J., Kononov, A., Queyranne, M., Sevastyanov, S., & Sviridenko, M. (2011). Integrality property in preemptive shop scheduling. Discrete Applied Mathematics, 159(5), 272–280. CrossRefGoogle Scholar
  3. Brucker, P., & Knust, S. (2000). Operations research: Complexity results of scheduling problems,
  4. Carlier, J., & Pinson, E. (1989). An algorithm for solving the job-shop problem. Management Science, 35(2), 164–176. CrossRefGoogle Scholar
  5. Chen, B., Potts, C., & Woeginger, G. (1998). A review of machine scheduling: complexity, algorithms and approximability. In Handbook of combinatorial optimization (Vol. 3, pp. 21–169). Boston: Kluwer Academic. Google Scholar
  6. Cole, R., Ost, K., & Schirra, S. (2001). Edge-coloring bipartite multigraphs in O(Elog D) time. Combinatorica, 21(1), 5–12. CrossRefGoogle Scholar
  7. Creignou, N., Khanna, S., & Sudan, M. (2001). Complexity classifications of Boolean constraint satisfaction problems. SIAM monographs on discrete mathematics and applications. Philadelphia: SIAM. CrossRefGoogle Scholar
  8. Drobouchevitch, I. G., & Strusevich, V. A. (1999). A polynomial algorithm for the three-machine open shop with a bottleneck machine. Annales of Operations Research, 92, 185–214. CrossRefGoogle Scholar
  9. Fisher, H., & Thompson, G. L. (1963). Probabilistic learning combinations of local job-shop scheduling rules. In J. F. Muth & G. L. Thompson (Eds.), Industrial scheduling (pp. 225–551). Englewood Cliffs: Prentice-Hall. Google Scholar
  10. Gabow, H., & Kariv, O. (1982). Algorithms for edge coloring bipartite graphs and multigraphs. SIAM Journal of Computing, 11, 117–129. CrossRefGoogle Scholar
  11. Garey, M., & Johnson, D. (1979). Computers and intractability. A guide to the theory of NP-completeness. A series of books in the mathematical sciences. San Francisco: Freeman. Google Scholar
  12. Gonzalez, T. (1979). A note on open shop preemptive schedules, Unit execution time shop problems. IEEE Transactions on Computing, 28, 782–786. CrossRefGoogle Scholar
  13. Gonzalez, T., & Sahni, S. (1976). Open shop scheduling to minimize finish time. Journal of the Association for Computing Machinery, 23(4), 665–679. CrossRefGoogle Scholar
  14. Hoogeveen, J., Lenstra, J. K., & Veltman, B. (1994). Three, four, five, six, or the complexity of scheduling with communication delays. Operations Research Letters, 16(3), 129–137. CrossRefGoogle Scholar
  15. Jackson, J. (1956). An extension of Jhonson’s results on job lot scheduling. Navel Research Logistics Quartely, 3(3), 201–203. CrossRefGoogle Scholar
  16. Jeavons, P., Cohen, D., & Gyssens, M. (1997). Closure properties of constraints. Journal of ACM, 44(4), 527–548. CrossRefGoogle Scholar
  17. Kashyrskikh, K. N., Sevastianov, S. V., & Tchernykh, I. D. (2000). Four-parametric complexity analysis of the open shop problem. Diskretnyi Analiz i Issledovanie Operatsii, Seriya 1, 7(4), 59–77 [in Russian]. Google Scholar
  18. Kashyrskikh, K. N., Kononov, A. V., Sevastianov, S. V., & Tchernykh, I. D. (2001). Polynomially solvable case of the 2-stage 3-machine open shop problem. Diskretnyi Analiz i Issledovanie Operatsii, Seriya 1, 8(1), 23–39 [in Russian]. Google Scholar
  19. König, D. (1916). Graphok és alkalmazásuk a determinánsok és a halmazok elméletére. Mathematikai és Természettudományi Értesitö, 34, 104–119. [German translation: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Mathematische Annalen, 77(1916), 453–465.] Google Scholar
  20. Kononov, A. V., & Sevastianov, S. V. (2000). On the complexity of the connected list vertex-coloring problem. Diskretnyi Analiz i Issledovanie Operatsii, Seriya 1, 7(2), 21–46 [in Russian]. Google Scholar
  21. Kononov, A., Sevastianov, S., & Sviridenko, M. (2009). Complete complexity classification of short shop scheduling. In A. Frid et al. (Eds.), Lecture notes in computer science: Vol. 5675. Computer science—theory and applications, 4th international computer science symposium in Russia, CSA 2009, Novosibirsk, Russia, August 2009 (pp. 227–236). Berlin: Springer. Proceedings Google Scholar
  22. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1993). Sequencing and scheduling: algorithms and complexity. In S. Graves, A. H. G. Rinnooy Kan, & P. Zipkin (Eds.), Logistics of production and inventory: Vol. 4. Handbooks in operations research and management science (pp. 445–522). Amsterdam: North-Holland. Google Scholar
  23. Lenstra, J. K., & Rinnooy Kan, A. H. G. (1978). Complexity of scheduling under precedence constraints. Operations Research, 26(1), 22–35. CrossRefGoogle Scholar
  24. Lenstra, J. K., Rinnooy Kan, A. H. G., & Brucker, P. (1977). Complexity of machine scheduling problems. Annales of Discrete Mathematics, 1, 343–362. CrossRefGoogle Scholar
  25. Lenstra, J., Shmoys, D., & Tardos, E. (1990). Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, Ser. A, 46(3), 259–271. CrossRefGoogle Scholar
  26. Leung, J. Y.-T. (Ed.) (2004). Handbook of scheduling. Algorithms, models, and performance analysis. Boca Raton: Chapman & Hall/CRC Computer and Information Science Series. Google Scholar
  27. Masuda, T., Ishii, H., & Nishida, T. (1985). The mixed shop scheduling problem. Discrete Applied Mathematics, 11(2), 175–186. CrossRefGoogle Scholar
  28. Melnikov, L., & Vizing, V. (1999). The edge chromatic number of a directed/mixed multigraph. Journal of Graph Theory, 31(4), 267–273. CrossRefGoogle Scholar
  29. Neumytov, J., & Sevastianov, S. (1993). An approximation algorithm with best possible bound for the counter routs problem with three machines. Upravlyaemye Sistemy, 31, 53–65 [in Russian]. Google Scholar
  30. Schaefer, T. J. (1978). The complexity of satisfiability problems. STOC, 1978, 216–226. Google Scholar
  31. Schrijver, A. (2003). Combinatorial optimization. Polyhedra and efficiency. Algorithms and combinatorics (Vol. 24B). Berlin: Springer. Google Scholar
  32. Sevastianov, S. (2005). An introduction to multi-parameter complexity analysis of discrete problems. European Journal of Operational Research, 165(2), 387–397. CrossRefGoogle Scholar
  33. Shakhlevich, N. V., Sotskov, Y. N., & Werner, F. (2000). Complexity of mixed shop scheduling problems: a survey. European Journal of Operational Research, 120, 343–351. CrossRefGoogle Scholar
  34. Timkovsky, V. G. (2003). Identical parallel machines vs. unit-time shops and preemptions vs. chains in scheduling complexity. European Journal of Operational Research, 149, 355–376. CrossRefGoogle Scholar
  35. Vizing, V. G. (1999). On a connected graph coloring in prescribed colors. Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 6(4), 36–43 [in Russian]. Google Scholar
  36. Vizing, V. G. (2002). A bipartite interpretation of a directed multigraph in problems of the coloring of incidentors. Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 9(1), 27–41 [in Russian]. Google Scholar
  37. Vizing, V. G. (2008). On the coloring of incidentors in a partially ordered multigraph. Diskretnyi Analiz i Issledovanie Operatsii, 15(1), 17–22 [in Russian]. Google Scholar
  38. Williamson, D., Hall, L., Hoogeveen, J., Hurkens, C., Lenstra, J. K., Sevastianov, S., & Shmoys, D. (1997). Short shop schedules. Operations Research, 45(2), 288–294. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Alexander Kononov
    • 1
  • Sergey Sevastyanov
    • 2
  • Maxim Sviridenko
    • 3
  1. 1.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  3. 3.IBM T.J. Watson Research CenterYorktown HeightsUSA

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