Time-Table Disjunctive Reasoning for the Cumulative Constraint

  • Steven GayEmail author
  • Renaud Hartert
  • Pierre Schaus
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9075)


Scheduling has been a successful domain of application for constraint programming since its beginnings. The cumulative constraint – which enforces the usage of a limited resource by several tasks – is one of the core components that are surely responsible of this success. Unfortunately, ensuring bound-consistency for the cumulative constraint is already NP-Hard. Therefore, several relaxations were proposed to reduce domains in polynomial time such as Time-Tabling, Edge-Finding, Energetic Reasoning, and Not-First-Not-Last. Recently, Vilim introduced the Time-Table Edge-Finding reasoning which strengthens Edge-Finding by considering the time-table of the resource. We pursue the idea of exploiting the time-table to detect disjunctive pairs of tasks dynamically during the search. This new type of filtering – which we call time-table disjunctive reasoning – is not dominated by existing filtering rules. We propose a simple algorithm that implements this filtering rule with a \(\mathcal {O}(n^2)\) time complexity (where \(n\) is the number of tasks) without relying on complex data structures. Our results on well known benchmarks highlight that using this new algorithm can substantially improve the solving process for some instances and only adds a marginally low computation overhead for the other ones.


Constraint programming Scheduling Cumulative constraint Time-table Disjunctive reasoning 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aggoun, A., Beldiceanu, N.: Extending chip in order to solve complex scheduling and placement problems. Mathematical and Computer Modelling 17(7), 57–73 (1993)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Baptiste, P., Le Pape, C.: Constraint propagation and decomposition techniques for highly disjunctive and highly cumulative project scheduling problems. Constraints 5(1–2), 119–139 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baptiste, P., Le Pape, C., Nuijten, W.: Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems, vol. 39. Springer (2001)Google Scholar
  4. 4.
    Beldiceanu, N., Carlsson, M.: A new multi-resource \(cumulatives\) constraint with negative heights. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 63–79. Springer, Heidelberg (2002) CrossRefGoogle 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, Heidelberg (2014) CrossRefGoogle Scholar
  6. 6.
    Kameugne, R., Fotso, L.P., Scott, J., Ngo-Kateu, Y.: A quadratic edge-finding filtering algorithm for cumulative resource constraints. Constraints 19(3), 243–269 (2014)MathSciNetGoogle Scholar
  7. 7.
    Kolisch, R., Schwindt, C., Sprecher, A.: Benchmark instances for project scheduling problems. In: Project Scheduling, pp. 197–212. Springer (1999)Google Scholar
  8. 8.
    Le Pape, C., Couronné, P., Vergamini, D., Gosselin, V.: Time-Versus-Capacity Compromises in Project Scheduling (1994)Google Scholar
  9. 9.
    Letort, A., Beldiceanu, N., Carlsson, M.: A scalable sweep algorithm for the cumulative constraint. In: Milano, M. (ed.) Principles and Practice of Constraint Programming. LNCS, pp. 439–454. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  10. 10.
    Lopez, P., Erschler, J., Esquirol, P.: Ordonnancement de tâches sous contraintes: une approche énergétique. Automatique-productique informatique industrielle 26(5–6), 453–481 (1992)zbMATHGoogle Scholar
  11. 11.
    Nuijten, W.P.M.: Time and resource constrained scheduling: a constraint satisfaction approach. PhD thesis, Technische Universiteit Eindhoven (1994)Google Scholar
  12. 12.
    OscaR Team. OscaR: Scala in OR (2012).
  13. 13.
    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) CrossRefGoogle Scholar
  14. 14.
    Schutt, A., Wolf, A.: A new 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)CrossRefGoogle Scholar
  15. 15.
    Vilím, P.: Edge finding filtering algorithm for discrete cumulative resources in O(kn log n). In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 802–816. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    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) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.ICTEAMUCLouvainLouvain-la-NeuveBelgium

Personalised recommendations