Advertisement

Approximation Schemes for a Class of Subset Selection Problems

  • Kirk Pruhs
  • Gerhard J. Woeginger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

In paper we develop an easily applicable algorithmic technique/tool for developing approximation schemes for certain types of combinatorial optimization problems. Special cases that are covered by our result show up in many places in the literature. For every such special case, a particular rounding trick has been implemented in a slightly different way, with slightly different arguments, and with slightly different worst case estimations. Usually, the rounding procedure depended on certain upper or lower bounds on the optimal objective value that have to be justified in a separate argument. Our easily applied result unifies many of these results, and sometimes it even leads to a simpler proof. We demonstrate how our result can be easily applied to a broad family of combinatorial optimization problems. As a special case, we derive the existence of an FPTAS for the scheduling problem of minimizing the weighted number of late jobs under release dates and preemption on a single machine. The approximability status of this problem has been open for some time.

Keywords

Schedule Problem Approximation Scheme Single Machine Knapsack Problem Combinatorial Optimization Problem 
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.
    Bellman, R.E., Dreyfus, S.E.: Applied Dynamic Programming. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  2. 2.
    Ergun, F., Sinha, R., Zhang, L.: An improved FPTAS for restricted shortest path. Information Processing Letters 83, 287–291 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W.H. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  4. 4.
    Gens, G.V., Levner, E.V.: Fast approximation algorithms for job sequencing with deadlines. Discrete Applied Mathematics 3, 313–318 (1981)zbMATHCrossRefGoogle Scholar
  5. 5.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17, 36–42 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Håstad, J.: Clique is hard to approximate within n1 − ε. Acta Mathematica 182, 105–142 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. Journal of the ACM 21, 277–292 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. Journal of the ACM 23, 317–327 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ibarra, O., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the ACM 22, 463–468 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–104. Plenum Press, New York (1972)Google Scholar
  12. 12.
    Lawler, E.L.: Fast approximation schemes for knapsack problems. Mathematics of Operations Research 4, 339–356 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lawler, E.L.: A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs. Annals of Operations Research 26, 125–133 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: Algorithms and complexity. In: Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, P.H. (eds.) Logistics of Production and Inventory, Handbooks in Operations Research and Management Science 4, pp. 445–522. North-Holland, Amsterdam (1993)CrossRefGoogle Scholar
  15. 15.
    Lawler, E.L., Moore, J.M.: A functional equation and its application to resource allocation and sequencing problems. Management Science 16, 77–84 (1969)zbMATHCrossRefGoogle Scholar
  16. 16.
    Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters 28, 213–219 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Martello, S., Toth, P.: Knapsack problems: Algorithms and computer implementations. John Wiley & Sons, England (1990)zbMATHGoogle Scholar
  18. 18.
    Sahni, S.: Algorithms for scheduling independent tasks. Journal of the ACM 23, 116–127 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Warburton, A.: Approximation of pareto optima in multiple-objective shortest path problems. Operations Research 35, 70–79 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS Journal on Computing 12, 57–75 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Kirk Pruhs
    • 1
  • Gerhard J. Woeginger
    • 2
  1. 1.Department of Computer ScienceUniversity of PittsburghUSA
  2. 2.Department of MathematicsUniversity of TwenteThe Netherlands

Personalised recommendations