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Packing: Scheduling, Embedding, and Approximating Metrics

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Computational Science and Its Applications – ICCSA 2004 (ICCSA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3045))

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Abstract

Many problems in computer science are related to scheduling problems or embedding problems. Therefore it is an interesting topic to find efficient (approximation) algorithms for these two classes of problems. In this paper, we present fast approximation algorithms for scheduling on unrelated machines, job shop scheduling, network embeddings and approximating metrics. As the usual technique, we consider the fractional relaxations of the integral problems. By appropriate formulation, these problems can be solved with the approximation algorithms for general packing problems by Jansen and Zhang [18] (with rounding techniques if necessary). For approximating metrics problem, which can not be solved by most traditional methods efficiently, we show that our algorithm can deliver the desired approximate solution fast.

This research was supported in part by the DFG Graduiertenkolleg 357, Effiziente Algorithmen und Mehrskalenmethoden, by EU Thematic Network APPOL, Approximation and Online Algorithms for Optimization Problems, IST-2001-32007, and by EU Project CRESCCO, Critical Resource Sharing for Cooperation in Complex Systems, IST-2001-33135.

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References

  1. Alon, N., Srinivasan, A.: Improved parallel approximation of a class of integer programming problems. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 562–573. Springer, Heidelberg (1996)

    Google Scholar 

  2. Applegate, D., Cook, W.: A computational study of the job-shop scheduling problem. ORSA Journal of Computing 3, 149–156 (1991)

    MATH  Google Scholar 

  3. Baltz, A., Srivastav, A.: Fast Approximation of Minimum Multicast Congestion – Implementation versus Theory. In: Proceedings of the 5th Conference on Algorithms and Complexity, CIAC 2003 (2003)

    Google Scholar 

  4. Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: Proceedings of the 37th IEEE Annual Symposium on Foundations of Computer Science, FOCS 1996, pp. 184–193 (1996)

    Google Scholar 

  5. Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC 1998 (1998)

    Google Scholar 

  6. Carlier, J., Pinson, E.: An algorithm for solving the job-shop problem. Management Science 35, 164–176 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Charikar, M., Chekuri, C., Goel, A., Guha, S.: Rounding via trees: deterministic approximation algorithms for group steiner trees and k-median. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, STOC 1998 (1998)

    Google Scholar 

  8. Charikar, M., Chekuri, C., Goel, A., Guha, S., Plotkin, S.: Approximating a finite metric by a small number of tree metrics. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, FOCS 1998, pp. 379–388 (1998)

    Google Scholar 

  9. Even, G., Naor, J.S., Rao, S., Schieber, B.: Fast approximate graph partitioning algorithms. SIAM. Journal on Computing 6, 2187–2214 (1999)

    Article  MathSciNet  Google Scholar 

  10. Garey, M., Johnson, D.: Computer and Intractability: A Guide to the Theory of NP-Completeness, W. W. H. Freeman and Company, NY (1979)

    Google Scholar 

  11. Garg, N., Könemann, J.: Fast and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of the 39th IEEE Annual Symposium on Foundations of Computer Science, FOCS 1998, pp. 300–309 (1998)

    Google Scholar 

  12. Goldberg, L.A., Paterson, M., Srinivasan, A., Sweedyk, E.: Better approximation guarantees for job-shop scheduling. SIAM Journal on Discrete Mathematics 14, 67–92 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grigoriadis, M.D., Khachiyan, L.G.: Coordination complexity of parallel pricedirective decomposition. Mathematics of Operations Research 2, 321–340 (1996)

    Article  MathSciNet  Google Scholar 

  14. Grigoriadis, M.D., Khachiyan, L.G.: Approximate minimum-cost multicommodity flows in O(ε−2knm) time. Mathematical Programming 75, 477–482 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Hu, T.C.: Optimum communication spanning trees. SIAM Journal on Computing 3, 188–195 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jansen, K.: Approximation algorithms for fractional covering and packing problems, and applications (2003) (manuscript)

    Google Scholar 

  17. Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 175–186. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  18. Jansen, K., Zhang, H.: Approximation algorithms for general packing problems with modified logarithmic potential function. In: Proceedings of 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002 (2002)

    Google Scholar 

  19. Jansen, K., Zhang, H.: An approximation algorithm for the multicast congestion problem via minimum Steiner trees. In: Proceedings of 3rd International Workshop on Approximation and Randomized Algorithms in Communication Networks, ARANCE 2002 (2002)

    Google Scholar 

  20. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: algorithms and complexity. In: Graves, S.C., et al. (eds.) Handbooks in Operations Research and Management Science: Logistics of Production and Inventory, vol. 4, pp. 445–522. Elsevier, Amsterdam (1993)

    Chapter  Google Scholar 

  21. Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science, FOCS 1998, pp. 422–431 (1988)

    Google Scholar 

  22. Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 24, 259–272 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Science 37, 130–143 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  25. Raghavan, P., Thompson, C.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmidt, J.P., Siegel, A., Srinivasan, A.: Chernoff-Hoeffding bounds for applications with limited independence. SIAM Journal on Discrete Mathematics 8, 223–250 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shmoys, D.B., Stein, C., Wein, J.: Improved approximation algorithms for shop scheduling problems. SIAM Journal on Computing 23, 617–632 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Vaidya, P.M.: Speeding up linear programming using fast matrix multiplication. In: Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, FOCS 1998, pp. 332–337 (1989)

    Google Scholar 

  29. Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. In: Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, FOCS 1989, pp. 338–343 (1989)

    Google Scholar 

  30. Villavicencio, J., Grigoriadis, M.D.: Approximate structured optimization by cyclic block-coordinate descent. In: Fischer, H., et al. (eds.) Applied Mathematics and Parallel Computing – Festschrift for Klaus Ritter, pp. 359–371. Physica-Verlag, Heidelberg (1996)

    Google Scholar 

  31. Villavicencio, J., Grigoriadis, M.D.: Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential. In: Pardalos, P., Hearn, D.W., Hager, W.W. (eds.) Network Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 450, pp. 471–485. Springer, Berlin (1997)

    Google Scholar 

  32. Williamson, D.P., Hall, L.A., Hoogeveen, J.A., Hurkens, C.A., Lenstra, J.K., Sevast’yanov, S.V., Shmoys, D.B.: Short shop schedules. Operations Research 45, 288–294 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wu, B.Y., Lancia, G., Bafna, V., Chao, K., Ravi, R., Tang, C.Y.: A polynomial time approximation scheme for minimum routing cost spanning trees. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1998 (1998)

    Google Scholar 

  34. Young, N.E.: Randomized rounding without solving the linear program. In: Proceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, SODA 1995, pp. 170–178 (1995)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science and Applied Mathematics, University of Kiel, Olshausenstraße 40, D-24098, Kiel, Germany

    Hu Zhang

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  1. Hu Zhang
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Editors and Affiliations

  1. Department of Chemistry, University of Perugia, Via Elce di Sotto, 8, I-06123, Perugia, Italy

    Antonio Laganá

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  3. William Norris Professor, Head of the Computer Science and Engineering Department, University of Minnesota, USA

    Vipin Kumar

  4. School of Computing, Soongsil University, Seoul, Korea

    Youngsong Mun

  5. OptimaNumerics Ltd., Cathedral House, 23-31 Waring Street, BT1 2DX, Belfast, UK

    C. J. Kenneth Tan

  6. Department of Mathematics and Computer Science, University of Perugia, via Vanvitelli, 1, I-06123, Perugia, Italy

    Osvaldo Gervasi

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© 2004 Springer-Verlag Berlin Heidelberg

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Zhang, H. (2004). Packing: Scheduling, Embedding, and Approximating Metrics. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds) Computational Science and Its Applications – ICCSA 2004. ICCSA 2004. Lecture Notes in Computer Science, vol 3045. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24767-8_81

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  • DOI: https://doi.org/10.1007/978-3-540-24767-8_81

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22057-2

  • Online ISBN: 978-3-540-24767-8

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