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International Conference on Integer Programming and Combinatorial Optimization

IPCO 1998: Integer Programming and Combinatorial Optimization pp 153–168Cite as

Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1412))

Abstract

The edge and vertex-disjoint path problems together with their unsplittable flow generalization are NP-hard problems with a multi- tude of applications in areas such as routing, scheduling and bin packing. Given the hardness of the problems, we study polynomial-time approxi- mation algorithms with bounded performance guarantees. We introduce techniques which yield new algorithms for a wide range of disjoint-path problems. We use two basic techniques. First, we propose simple greedy algorithms for edge- and vertex-disjoint paths and second, we propose the use of a framework based on packing integer programs for more general problems such as unsplittable flow. As part of our tools we develop im- proved approximation algorithms for a class of packing integer programs, a result that we believe is of independent interest.

Research partly supported by NSF Award CCR-9308701 and NSF Career Award CCR-9624828.

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References

  1. R. Aharoni, P. Erdös, and N. Linial. Optima of dual integer linear programs. Comb., 8:13–20, 1988.

    MATH  Google Scholar 

  2. Y. Aumann and Y. Rabani. Improved bounds for all-optical routing. In Proc. 6th ACM-SIAM Symp. on Discrete Algorithms, pages 567–576, 1995.

    Google Scholar 

  3. A. Z. Broder, A. M. Frieze, and E. Upfal. Static and dynamic path selection on expander graphs: a random walk approach. In Proc. 29th Ann. ACM Symp. on Theory of Computing, 531–539, 1997.

    Google Scholar 

  4. C. Cooper. The thresold of hamilton cycles in the square of a random graph. Random Structures and Algorithms, 5:25–31, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Fleischner. The square of every two-connected graph is hamiltonian. J. of Combinatorial Theory B, 16:29–34, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Frank. Packing paths, cuts and circuits — A survey. In B. Korte, L. Lovász, H. J. Prömel, and A. Schrijver, editors, Paths, Flows and VLSI-Layout, pages 49–100. Springer-Verlag, Berlin, 1990.

    Google Scholar 

  7. N. Garg, V. Vazirani, and M. Yannakakis. Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica, 18:3–20, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. M. Karp, F. T. Leighton, R. L. Rivest, C. D. Thompson, U. V. Vazirani, and V. V. Vazirani. Global wire routing in two-dimensional arrays. Algorithmica, 2:113–129, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. M. Karp. On the computational complexity of combinatorial problems. Networks, 5:45–68, 1975.

    MATH  MathSciNet  Google Scholar 

  10. J. M. Kleinberg. Approximation algorithms for disjoint paths problems. PhD thesis, MIT, Cambridge, MA, May 1996.

    Google Scholar 

  11. J. M. Kleinberg. Single-source unsplittable flow. In Proc. 37th Ann. Symp. on Foundations of Computer Science, pages 68–77, October 1996.

    Google Scholar 

  12. J. M. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In Proc. 37th Ann. Symp. on Foundations of Computer Science, pages 86–95, 1996.

    Google Scholar 

  13. J. M. Kleinberg and É. Tardos. Disjoint paths in densely-embedded graphs. In Proc. 36th Ann. Symp. on Foundations of Computer Science, pages 52–61, 1995.

    Google Scholar 

  14. S. G. Kolliopoulos and C. Stein. Improved approximation algorithms for unsplittable flow problems. In Proc. 38th Ann. Symp. on Foundations of Computer Science, pages 426–435, 1997.

    Google Scholar 

  15. S. G. Kolliopoulos and C. Stein. Approximating disjoint-path problems using greedy algorithms and packing integer programs. Technical Report PCS TR97-325, Department of Computer Science, Dartmouth College, 1997.

    Google Scholar 

  16. F. T. Leighton and S. B. Rao. Circuit switching: A multi-commodity flow approach. In Workshop on Randomized Parallel Computing, 1996.

    Google Scholar 

  17. T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In Proc. 29th Ann. Symp. on Foundations of Computer Science, pages 422–431, 1988.

    Google Scholar 

  18. Y.-L. Lin and S. E. Skiena. Algorithms for square roots of graphs. SIAM J. on Discrete Mathematics, 8(1):99–118, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. Lovász. On the ratio of optimal and fractional covers. Discrete Mathematics, 13:383–390, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Martin and D. B. Shmoys. A new approach to computing optimal schedules for the job-shop scheduling problem. In Proc. 5th Conference on Integer Programming and Combinatorial Optimization, pages 389–403, 1996.

    Google Scholar 

  21. D. Peleg and E. Upfal. Disjoint paths on expander graphs. Comb., 9:289–313, 1989.

    MATH  MathSciNet  Google Scholar 

  22. S. Plotkin. Competitive routing of virtual circuits in ATM networks. IEEE J. Selected Areas in Comm., 1128–1136, 1995.

    Google Scholar 

  23. S. Plotkin, D. B. Shmoys, and E. Tardos. Fast approximation algorithms for fractional packing and covering problems. Math. of Oper. Res., 20:257–301, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  24. P. Raghavan. Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. of Computer and System Sciences, 37:130–143, 1988.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  26. N. Robertson and P. D. Seymour. Outline of a disjoint paths algorithm. In B. Korte, L. Lovász, H. J. Prömel, and A. Schrijver, editors, Paths, Flows and VLSI-Layout. Springer-Verlag, Berlin, 1990.

    Google Scholar 

  27. A. Schrijver. Homotopic routing methods. In B. Korte, L. Lovász, H. J. Prömel, and A. Schrijver, editors, Paths, Flows and VLSI-Layout. Springer, Berlin, 1990.

    Google Scholar 

  28. D. B. Shmoys, C. Stein, and J. Wein. Improved approximation algorithms for shop scheduling problems. SIAM J. on Computing, 23(3):617–632, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  29. A. Srinivasan. Improved approximations of packing and covering problems. In Proc. 27th Ann. ACM Symp. on Theory of Computing, pages 268–276, 1995.

    Google Scholar 

  30. A. Srinivasan. An extension of the Lovász Local Lemma and its applications to integer programming. In Proc. 7th ACM-SIAM Symp. on Discrete Algorithms, pages 6–15, 1996.

    Google Scholar 

  31. A. Srinivasan. Improved approximations for edge-disjoint paths, unsplittable flow and related routing problems. In Proc. 38th Ann. Symp. on Foundations of Computer Science, pages 416–425, 1997.

    Google Scholar 

  32. A. Srinivasan and C.-P. Teo. A constant-factor approximation algorithm for packet routing and balancing local vs. global criteria. In Proc. 29th Ann. ACM Symp. on Theory of Computing, pages 636–643, 1997.

    Google Scholar 

  33. C. Stein. Approximation Algorithms for Multicommodity Flow and Shop Scheduling Problems. PhD thesis, MIT, Cambridge, MA, August 1992.

    Google Scholar 

  34. P. Underground. On graphs with hamiltonian squares. Disc. Math., 21:323, 1978.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Dartmouth College, Hanover, NH, 03755-3510, USA

    Stavros G. Kolliopoulos & Clifford Stein

Authors
  1. Stavros G. Kolliopoulos
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  2. Clifford Stein
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Editor information

Editors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6020 Annapolis, Houston, TX, 77005, USA

    Robert E. Bixby

  2. PROS Strategic Solutions, 3223 Smith Street, Houston, TX, 77006, USA

    E. Andrew Boyd

  3. Department of Industrial, Engineering, Texas A&M University, 1000 Country Place Dr. Apt. 69, Houston, TX, 77079, USA

    Roger Z. Ríos-Mercado

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© 1998 Springer-Verlag Berlin Heidelberg

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Kolliopoulos, S.G., Stein, C. (1998). Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_12

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  • DOI: https://doi.org/10.1007/3-540-69346-7_12

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  • Print ISBN: 978-3-540-64590-0

  • Online ISBN: 978-3-540-69346-8

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