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International Colloquium on Automata, Languages, and Programming

ICALP 2005: Automata, Languages and Programming pp 828–839Cite as

A Faster Combinatorial Approximation Algorithm for Scheduling Unrelated Parallel Machines

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3580))

Abstract

We consider the problem of scheduling n independent jobs on m unrelated parallel machines without preemption. Job i takes processing time p ij on machine j, and the total time used by a machine is the sum of the processing times for the jobs assigned to it. The objective is to minimize makespan. The best known approximation algorithms for this problem compute an optimum fractional solution and then use rounding techniques to get an integral 2-approximation.

In this paper we present a combinatorial approximation algorithm that matches this approximation quality. It is much simpler than the previously known algorithms and its running time is better. This is the first time that a combinatorial algorithm always beats the interior point approach for this problem. Our algorithm is a generic minimum cost flow algorithm, without any complex enhancements, tailored to handle unsplittable flow. It pushes unsplittable jobs through a two-layered bipartite generalized network defined by the scheduling problem. In our analysis, we take advantage from addressing the approximation problem directly. In particular, we replace the classical technique of solving the LP-relaxation and rounding afterwards by a completely integral approach. We feel that this approach will be helpful also for other applications.

This work has been partially supported by the DFG-Sonderforschungsbereich 376 Massive Parallelität: Algorithmen, Entwurfsmethoden, Anwendungen, by the European Union within the 6th Framework Programme under contract 001907 (DELIS) and by the DFG Research Training Group GK-693 of the Paderborn Institute for Scientific Computation (PaSCo)

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  3. Dinitz, Y., Garg, N., Goemans, M.X.: On the single-source unsplittable flow problem. Combinatorica 19(1), 17–41 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fleischer, L., Wayne, K.D.: Fast and simple approximation schemes for generalized flow. Mathematical Programming 91(2), 215–238 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: Computing Nash equilibria for scheduling on restricted parallel links. In: Proceedings of the 36th Annual ACM Symposium on the Theory of Computing (STOC 2004), pp. 613–622 (2004)

    Google Scholar 

  6. Goldberg, A.V., Plotkin, S.A., Tardos, E.: Combinatorial algorithms for the generalized circulation problem. Math. of Operations Research 16, 351–379 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Goldfarb, D., Jin, Z., Orlin, J.B.: Polynomial-time highest-gain augmenting path algorithms for the generalized circulation problem. Math. of Operations Research 22, 793–802 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gondran, M., Minoux, M.: Graphs and Algorithms. Wiley, Chichester (1984)

    MATH  Google Scholar 

  9. Graham, R.L.: Bounds for certain multiprocessor anomalies. Bell System Technical Journal 45, 1563–1581 (1966)

    Google Scholar 

  10. Hochbaum, D.S.: Approximation Algorithms for NP-hard Problems. PWS Publishing Co. (1996)

    Google Scholar 

  11. Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. Journal of the ACM 23(2), 317–327 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jansen, K., Porkolab, L.: Improved approximation schemes for scheduling unrelated parallel machines. Math. of Operations Research 26(2), 324–338 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jewell, W.S.: Optimal flow through networks with gains. Operations Research 10, 476–499 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kapoor, S., Vaidya, P.M.: Fast algorithms for convex quadratic programming and multicommodity flows. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC 1986), pp. 147–159 (1986)

    Google Scholar 

  15. Kleinberg, J.: Single-source unsplittable flow. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science (FOCS 1996), pp. 68–77 (1996)

    Google Scholar 

  16. Kolliopoulos, S.G., Stein, C.: Approximation algorithms for single-source unsplittable flow. SIAM Journal on Computing 31, 919–946 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mokotoff, E., Chrétienne, P.: A cutting plane algorithm for the unrelated parallel machine scheduling problem. European Journal of Operational Research 141, 515–525 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Onaga, K.: Dynamic programming of optimum flows in lossy communication nets. IEEE Transactions on Circuit Theory 13, 308–327 (1966)

    Google Scholar 

  20. Plotkin, S.A., Shmoys, D.B., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Math. of Operations Research 20(2), 257–301 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Radzik, T.: Faster algorithms for the generalized network flow problem. Math. of Operations Research 23, 69–100 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Radzik, T.: Improving time bounds on maximum generalised flow computations by contracting the network. Theoretical Computer Science 312(1), 75–97 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shmoys, D.B., Tardos, E.: An approximation algorithm for the generalized assignment problem. Mathematical Programming 62, 461–474 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sourd, F.: Scheduling tasks on unrelated machines: Large neighborhood improvement procedures. Journal of Heuristics 7, 519–531 (2001)

    Article  MATH  Google Scholar 

  25. Tardos, E., Wayne, K.D.: Simple generalized maximum flow algorithms. In: Proceedings of the 6th Integer Programming and Combinatorial Optimization Conference (IPCO 1998), pp. 310–324 (1998)

    Google Scholar 

  26. Truemper, K.: On max flows with gains and pure min-cost flows. SIAM Journal on Applied Mathematics 32(2), 450–456 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  27. Vaidya, P.M.: Speeding up linear programming using fast matrix multiplication. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS 1989), pp. 332–337 (1989)

    Google Scholar 

  28. van de Velde, S.L.: Duality-based algorithms for scheduling unrelated parallel machines. ORSA Journal on Computing 5(2), 182–205 (1993)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Martin Gairing, Burkhard Monien & Andreas Woclaw

Authors
  1. Martin Gairing
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  2. Burkhard Monien
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  3. Andreas Woclaw
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Editor information

Editors and Affiliations

  1. CITI / Departamento de Informática, Universidade Nova de Lisboa, Portugal

    Luís Caires

  2. Dipartimento di Informatica, Sistemi e Produzione, Università di Roma “Tor Vergata”, via del Politecnico 1, 00133, Roma, Italy

    Giuseppe F. Italiano

  3. Departamento de Informatica, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal

    Luís Monteiro

  4. Ecole Polytechnique, Rue de Saclay, 91128, Palaiseau Cedex, France

    Catuscia Palamidessi

  5. Computer Science Department, Google Inc. and Columbia University, 1214 Amsterdam Avenue, NY 10027, New York, USA

    Moti Yung

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© 2005 Springer-Verlag Berlin Heidelberg

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Gairing, M., Monien, B., Woclaw, A. (2005). A Faster Combinatorial Approximation Algorithm for Scheduling Unrelated Parallel Machines. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_67

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  • DOI: https://doi.org/10.1007/11523468_67

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-27580-0

  • Online ISBN: 978-3-540-31691-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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