Journal of Scheduling

, Volume 14, Issue 5, pp 501–509 | Cite as

Isomorphic coupled-task scheduling problem with compatibility constraints on a single processor

  • G. SimoninEmail author
  • B. Darties
  • R. Giroudeau
  • J.-C. König


The problem presented in this paper is a generalization of the usual coupled-tasks scheduling problem in presence of compatibility constraints. The reason behind this study is the data acquisition problem for a submarine torpedo. We investigate a particular configuration for coupled tasks (any task is divided into two sub-tasks separated by an idle time), in which the idle time of a coupled task is equal to the sum of durations of its two sub-tasks. We prove \(\mathcal{NP}\)-completeness of the minimization of the schedule length, we show that finding a solution to our problem amounts to solving a graph problem, which in itself is close to the minimum-disjoint-path cover (min-DCP) problem. We design a \((\frac{3a+2b}{2a+2b})\)-approximation, where a and b (the processing time of the two sub-tasks) are two input data such as a>b>0, and that leads to a ratio between \(\frac {3}{2}\) and \(\frac{5}{4}\). Using a polynomial-time algorithm developed for some class of graph of min-DCP, we show that the ratio decreases to \(\frac{1+\sqrt{3}}{2}\approx 1.37\).


Coupled-tasks Complexity Compatibility graph Polynomial-time approximation 


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  1. Ahr, D., Békési, J., Galambos, G., Oswald, M., & Reinelt, G. (2004). An exact algorithm for scheduling identical coupled-tasks. Mathematical Methods of Operations Research, 59(11), 193–203. Google Scholar
  2. Arikati, R. S., & Rangan, P. C. (1990). Linear algorithm for optimal path cover problem on interval graphs. Information Processing Letters, 35(3), 149–153. CrossRefGoogle Scholar
  3. Berman, P., & Karpinski, M. (2006). 8/7-approximation algorithm for (1,2)-TSP. In SODA ’06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm (pp. 641–648). New York: ACM. CrossRefGoogle Scholar
  4. Blażewicz, J., Ecker, K., Kis, T., Potts, C. N., Tanas, M., & Whitehead, J. (2009). Scheduling of coupled tasks with unit processing times. Technical report, Poznan University of Technology. Google Scholar
  5. Boesch, F. T., Chen, S., & McHugh, B. (1974). On covering the points of a graph with point-disjoint paths. Graphs and Combinatorics, 406, 201–212. CrossRefGoogle Scholar
  6. Chrétienne, Ph., & Picouleau, C. (1995). Scheduling with communication delays: a survey. In Scheduling theory and its applications (pp. 641–648). New York: Wiley. Google Scholar
  7. Detti, P., & Meloni, C. (2004). A linear algorithm for the hamiltonian completion number of the line graph of a cactus. Discrete Applied Mathematics, 136(2–3), 197–215. CrossRefGoogle Scholar
  8. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: Freeman. Google Scholar
  9. Goodman, S. E., Hedetniemi, S. T., & Slater, P. J. (1975). Advances on the hamiltonian completion problem. Journal of the ACM, 22(3), 352–360. CrossRefGoogle Scholar
  10. Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics, 5, 287–326. CrossRefGoogle Scholar
  11. Hung, R.-W., & Chang, M.-S. (2006). Solving the path cover problem on circular-arc graphs by using an approximation algorithm. Discrete Applied Mathematics, 154(1), 76–105. CrossRefGoogle Scholar
  12. Hung, R.-W., & Chang, M.-S. (2007). Finding a minimum path cover of a distance-hereditary graph in polynomial time. Discrete Applied Mathematics, 155(17), 2242–2256. CrossRefGoogle Scholar
  13. Kundu, S. (1976). A linear algorithm for the hamiltonian completion number of a tree. Information Processing Letters, 5, 55–57. CrossRefGoogle Scholar
  14. Lehoux-Lebacque, V., Brauner, N., & Finke, G. (2009). Identical coupled task scheduling: polynomial complexity of the cyclic case. Les Cahiers Leibniz, 179. Google Scholar
  15. Moran, S., & Wolfstahl, Y. (1988). Optimal covering of cacti by vertex-disjoint paths. Theoretical Computer Science, 84, 179–197. CrossRefGoogle Scholar
  16. Nakano, K., Olariu, S., & Zomaya, A. Y. (2003). A time-optimal solution for the path cover problem on cographs. Theoretical Computer Science, 290(3), 1541–1556. CrossRefGoogle Scholar
  17. Orman, A. J., & Potts, C. N. (1997). On the complexity of coupled-task scheduling. Discrete Applied Mathematics, 72, 141–154. CrossRefGoogle Scholar
  18. Schrijver, A. (2004). Combinatorial optimization: polyhedra and efficiency (algorithms and combinatorics). Berlin: Springer. Google Scholar
  19. Shapiro, R. D. (1980). Scheduling coupled tasks. Naval Research Logistics Quarterly, 27, 477–481. CrossRefGoogle Scholar
  20. Simonin, G. (2009). Impact du graphe de compatibilité sur la complexité et l’approximation des problèmes d’ordonnancement en présence de tâches-couplées. Ph.D thesis, LIRMM, December 2009. Google Scholar
  21. Simonin, G., Giroudeau, R., & König, J.-C. (2009a). Complexity and approximation for scheduling problem for a torpedo. In CIE’39: The 39th international conference on computers and industrial engineering, IEEE, Troyes, France (pp. 300–304). Google Scholar
  22. Simonin, G., Giroudeau, R., & König, J.-C. (2009b). Extended matching problem for a coupled-tasks scheduling problem. In TMFCS’09: international conference on theoretical and mathematical foundations of computer science, Orlando, Florida (pp. 082–089). Google Scholar
  23. Srikant, R., Sundaram, R., Sher Singh, K., & Rangan, P. C. (1993). Optimal path cover problem on block graphs and bipartite permutation graphs. Theoretical Computer Science, 115(2), 351–357. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • G. Simonin
    • 1
    Email author
  • B. Darties
    • 2
  • R. Giroudeau
    • 1
  • J.-C. König
    • 1
  1. 1.LIRMM UMR 5506Montpellier Cedex 5France
  2. 2.LE2I UMR 5158DijonFrance

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