Improving Energetic Propagations for Cumulative Scheduling

  • Alexander TeschEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)


We consider the Cumulative Scheduling Problem (CuSP) in which a set of n jobs must be scheduled according to release dates, due dates and cumulative resource constraints. In constraint programming, the CuSP is modeled as the cumulative constraint. Among the most common propagation algorithms for the CuSP there is energetic reasoning [1] with a complexity of \(O(n^3)\) and edge-finding [21] with \(O(kn \log n)\) where \(k \le n\) is the number of different resource demands. We consider the complete versions of the propagators that perform all deductions in one call of the algorithm. In this paper, we introduce the energetic edge-finding rule that is a generalization of both energetic reasoning and edge-finding. Our main result is a complete energetic edge-finding algorithm with a complexity of \(O(n^2 \log n)\) which improves upon the complexity of energetic reasoning. Moreover, we show that a relaxation of energetic edge-finding with a complexity of \(O(n^2)\) subsumes edge-finding while performing stronger propagations from energetic reasoning. A further result shows that energetic edge-finding reaches its fixpoint in strongly polynomial time. Our main insight is that energetic schedules can be interpreted as a single machine scheduling problem from which we deduce a monotonicity property that is exploited in the algorithms. Hence, our algorithms improve upon the strength and the complexity of energetic reasoning and edge-finding whose complexity status seemed widely untouchable for the last decades.



The author would like to thank anonymous reviewers for their helpful comments on the paper, especially for the advice to take a simpler representation of \(\tilde{\omega }(t)\) as given in the current version of the paper.


  1. 1.
    Baptiste, P., Le Pape, C., Nuijten, W.: Satisfiability tests and time bound adjustments for cumulative scheduling problems. Ann. Oper. Res. 92, 305–333 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berthold, T., Heinz, S., Schulz, J.: An approximative criterion for the potential of energetic reasoning. In: Marchetti-Spaccamela, A., Segal, M. (eds.) TAPAS 2011. LNCS, vol. 6595, pp. 229–239. Springer, Heidelberg (2011). Scholar
  3. 3.
    Bonifas, N.: A \(O(n^2 log(n))\) propagation for the Energy Reasoning, Conference Paper, Roadef 2016. (2016)Google Scholar
  4. 4.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.C.: Computational Geometry. In: de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.C. (eds.) Computational Geometry. Springer, Heidelberg (2000). Scholar
  5. 5.
    Derrien, A., Petit, T.: A new characterization of relevant intervals for energetic reasoning. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 289–297. Springer, Cham (2014). Scholar
  6. 6.
    Erschler, J., Lopez, P.: Energy-based approach for task scheduling under time and resources constraints. In: 2nd International Workshop on Project Management and Scheduling, pp. 115–121 (1990)Google Scholar
  7. 7.
    Gay, S., Hartert, R., Schaus, P.: Simple and scalable time-table filtering for the cumulative constraint. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 149–157. Springer, Cham (2015). Scholar
  8. 8.
    Goemans, M.X.: A supermodular relaxation for scheduling with release dates. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 288–300. Springer, Heidelberg (1996). Scholar
  9. 9.
    Hooker, J.N.: A hybrid method for planning and scheduling. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 305–316. Springer, Heidelberg (2004). Scholar
  10. 10.
    Kameugne, R., Fetgo, S.B., Fotso, L.P.: Energetic extended edge finding filtering algorithm for cumulative resource constraints. Am. J. Oper. Res. 3(06), 589 (2013)CrossRefGoogle Scholar
  11. 11.
    Lenstra, J.K., Kan, A.R., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977)Google Scholar
  12. 12.
    Letort, A., Beldiceanu, N., Carlsson, M.: A scalable sweep algorithm for the cumulative constraint. In: Milano, M. (ed.) CP 2012. LNCS, pp. 439–454. Springer, Heidelberg (2012). Scholar
  13. 13.
    Kolisch, R., Sprecher, A.: PSPLIB-a project scheduling problem library: OR software-ORSEP operations research software exchange program. Eur. J. Oper. Res. 96(1), 205–216 (1997)CrossRefGoogle Scholar
  14. 14.
    Mercier, L., Van Hentenryck, P.: Edge finding for cumulative scheduling. INFORMS J. Comput. 20(1), 143–153 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mercier, L., Van Hentenryck, P.: Strong polynomiality of resource constraint propagation. Discrete Optim. 4(3–4), 288–314 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ouellet, P., Quimper, C.-G.: Time-table extended-edge-finding for the cumulative constraint. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 562–577. Springer, Heidelberg (2013). Scholar
  17. 17.
    Schutt, A., Feydy, T., Stuckey, P.J., Wallace, M.G.: Explaining the cumulative propagator. Constraints 16(3), 250–282 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schutt, A., Wolf, A.: A new \({\cal{O}}(n^2\log n)\) not-first/not-last pruning algorithm for cumulative resource constraints. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 445–459. Springer, Heidelberg (2010). Scholar
  19. 19.
    Schutt, A., Feydy, T., Stuckey, P.J.: Explaining time-table-edge-finding propagation for the cumulative resource constraint. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 234–250. Springer, Heidelberg (2013). Scholar
  20. 20.
    Tesch, A.: A nearly exact propagation algorithm for energetic reasoning in \(\cal{O}(n^2 \log n)\). In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 493–519. Springer, Cham (2016). Scholar
  21. 21.
    Vilím, P.: Edge finding filtering algorithm for discrete cumulative resources in \({\cal{O}}(kn {\rm log} n)\). In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 802–816. Springer, Heidelberg (2009). Scholar
  22. 22.
    Vilím, P.: Timetable edge finding filtering algorithm for discrete cumulative resources. In: Achterberg, T., Beck, J.C. (eds.) CPAIOR 2011. LNCS, vol. 6697, pp. 230–245. Springer, Heidelberg (2011). Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Zuse Institute Berlin (ZIB)BerlinGermany

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