Advertisement

Abstract

This paper investigates the relationship between the dimension theory of partial orders and the problem of scheduling precedence-constrained jobs on a single machine to minimize the weighted completion time. Surprisingly, we show that the vertex cover graph associated to the scheduling problem is exactly the graph of incomparable pairs defined in dimension theory. This equivalence gives new insights on the structure of the problem and allows us to benefit from known results in dimension theory. In particular, the vertex cover graph associated to the scheduling problem can be colored efficiently with at most k colors whenever the associated poset admits a polynomial time computable k-realizer. Based on this approach, we derive new and better approximation algorithms for special classes of precedence constraints, including convex bipartite and semi-orders, for which we give \((1+\frac{1}{3})\)-approximation algorithms. Our technique also generalizes to a richer class of posets obtained by lexicographic sum.

Keywords

Schedule Problem Approximation Algorithm Polynomial Time Partial Order Single Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambühl, C., Mastrolilli, M.: Single machine precedence constrained scheduling is a vertex cover problem. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 28–39. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Chekuri, C., Motwani, R.: Precedence constrained scheduling to minimize sum of weighted completion times on a single machine. Discrete Applied Mathematics 98(1-2), 29–38 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chudak, F.A., Hochbaum, D.S.: A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Operations Research Letters 25, 199–204 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Correa, J.R., Schulz, A.S.: Single machine scheduling with precedence constraints. IPCO 2004 30(4), 1005–1021 (2005); Extended abstract in Proceedings of the 10th Conference on Integer Programming and Combinatorial Optimization (IPCO 2004), pp. 283–297 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dushnik, B., Miller, E.: Partially ordered sets. American Journal of Mathematics 63, 600–610 (1941)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Felsner, S., Möhring, R.: Semi order dimension two is a comparability invariant. Order (15), 385–390 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Felsner, S., Trotter, W.T.: Dimension, graph and hypergraph coloring. Order 17(2), 167–177 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing 31(2), 601–625 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Graham, R., Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. In: Annals of Discrete Mathematics, vol. 5, pp. 287–326. North–Holland, Amsterdam (1979)Google Scholar
  10. 10.
    Hall, L.A., Schulz, A.S., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line algorithms. Mathematics of Operations Research 22, 513–544 (1997)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kolliopoulos, S.G., Steiner, G.: Partially-ordered knapsack and applications to scheduling. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 612–624. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Lawler, E.L.: Sequencing jobs to minimize total weighted completion time subject to precedence constraints. Annals of Discrete Mathematics 2, 75–90 (1978)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lenstra, J.K., Rinnooy Kan, A.H.G.: The complexity of scheduling under precedence constraints. Operations Research 26, 22–35 (1978)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Margot, F., Queyranne, M., Wang, Y.: Decompositions, network flows and a precedence constrained single machine scheduling problem. Operations Research 51(6), 981–992 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Möhring, R.H.: Computationally tractable classes of ordered sets. In: Rival, I. (ed.) Algorithms and Order, pp. 105–193. Kluwer Academic, Dordrecht (1989)Google Scholar
  17. 17.
    Pisaruk, N.N.: A fully combinatorial 2-approximation algorithm for precedence-constrained scheduling a single machine to minimize average weighted completion time. Discrete Applied Mathematics 131(3), 655–663 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Potts, C.N.: An algorithm for the single machine sequencing problem with precedence constraints. Mathematical Programming Study 13, 78–87 (1980)MATHMathSciNetGoogle Scholar
  19. 19.
    Rabinovitch, I.: The dimension of semiorders. Journal of Combinatorial Theory Series A(25), 50–61 (1978)MathSciNetGoogle Scholar
  20. 20.
    Schulz, A.S.: Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 301–315. Springer, Heidelberg (1996)Google Scholar
  21. 21.
    Schuurman, P., Woeginger, G.J.: Polynomial time approximation algorithms for machine scheduling: ten open problems. Journal of Scheduling 2(5), 203–213 (1999)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins Series in the Mathematical Sciences. The Johns Hopkins University Press (1992)Google Scholar
  23. 23.
    Trotter, W.T.: New perspectives on interval orders and interval graphs. In: Bailey, R.A. (ed.) Surveys in Combinatorics, London. Mathematical Society Lecture Note Series, vol. 241, pp. 237–286 (1997)Google Scholar
  24. 24.
    Woeginger, G.J.: On the approximability of average completion time scheduling under precedence constraints. Discrete Applied Mathematics 131(1), 237–252 (2003)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Yannakakis, M.: On the complexity of partial order dimension problem. SIAM Journal on Algebraic and Discrete Methods 22(3), 351–358 (1982)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christoph Ambühl
    • 1
  • Monaldo Mastrolilli
    • 2
  • Ola Svensson
    • 2
  1. 1.University of LiverpoolGreat Britain
  2. 2.IDSIASwitzerland

Personalised recommendations