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Using linear programming in the design and analysis of approximation algorithms: Two illustrative problems

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Approximation Algorithms for Combinatiorial Optimization (APPROX 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1444))

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Abstract

One of the foremost techniques in the design and analysis of approximation algorithms is to round the optimal solution to a linear programming relaxation in order to compute a near-optimal solution to the problem at hand. We shall survey recent work in this vein for two particular problems: the uncapacitated facility location problem and the problem of scheduling precedence-constrained jobs on one machine so as to minimize a weighted average of their completion times.

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References

  1. M. L. Balinksi. On finding integer solutions to linear programs. In Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, pages 225–248. IBM, 1966.

    Google Scholar 

  2. C. Chekuri and R. Motwani. Precedence constrained scheduling to minimize weighted completion time on a single machine. Unpublished manuscript, 1997.

    Google Scholar 

  3. C. Chekuri, R. Motwani, B. Natarajan, and C. Stein. Approximation techniques for average completion time scheduling. Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 609–618, 1997.

    Google Scholar 

  4. F. A. Chudak. Improved approximation algorithms for uncapacitated facility location. In: Proceedings of the 6th Integer Programming and Combinatorial Optimization Conference (IPCO), 1998, to appear.

    Google Scholar 

  5. F. A. Chudak and D. S. Hochbaum. A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Unpublished manuscript, 1997.

    Google Scholar 

  6. F. A. Chudak and D. B Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. Unpublished manuscript, 1997.

    Google Scholar 

  7. V. Chvátal. A greedy heuristic for the set covering problem. Math. Oper. Res., 4:233–235, 1979.

    MATH  MathSciNet  Google Scholar 

  8. M. E. Dyer and L. A. Wolsey. Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Appl. Math., 26:255–270, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Even, J. Naor, S. Rao, and B. Schieber. Divide-and-conquer approximation algorithms via spreading metrics. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science, pages 62–71, 1995.

    Google Scholar 

  10. M. X. Goemans. Personal communication, June, 1996.

    Google Scholar 

  11. R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math., 5:287–326, 1979.

    MATH  MathSciNet  Google Scholar 

  12. S. Guha and S. Khuller. Greedy strikes back: Improved facility location algorithms. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 649–657, 1998.

    Google Scholar 

  13. L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling to minimize the average completion time: on-line and off-line approximation algorithms. Math. Oper. Res., 22:513–544, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. A. Hall, D. B. Shmoys, and J. Wein. Scheduling to minimize the average completion time: on-line and off-line algorithms. In Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 142–151, 1996.

    Google Scholar 

  15. D. S. Hochbaum. Heuristics for the fixed cost median problem. Math. Programming, 22:148–162, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  16. D. S. Hochbaum, editor. Approximation algorithms for NP-hard problems, Boston, MA, 1997. PWS.

    Google Scholar 

  17. D. S. Hochbaum, N. Megiddo, J. Naor, and A. Tamir. Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Math. Programming, 62:69–83, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. S. Hochbaum and D. B. Shmoys. A best possible approximation algorithm for the k-center problem. Math. Oper. Res., 10:180–184, 1985.

    MATH  MathSciNet  Google Scholar 

  19. D. S. Johnson. Approximation algorithms for combinatorial problems. J. Comput. System Sci., 9:256–278, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. A. Kuehn and M. J. Hamburger. A heuristic program for locating warehouses. Management Sci., 9:643–666, 1963.

    Google Scholar 

  21. E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, New York, 1976.

    MATH  Google Scholar 

  22. J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker. Complexity of machine scheduling problems. Ann. Discrete Math., 1:343–362, 1977.

    Article  MATH  Google Scholar 

  23. J.-H. Lin and J. S. Vitter. Approximation algorithms for geometric median problems. Inform. Proc. Lett., 44:245–249, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.-H. Lin and J. S. Vitter. ε-approximations with minimum packing constraint violation. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 771–782, 1992.

    Google Scholar 

  25. L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Math., 13:383–390, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  26. A. S. Manne. Plant location under economies-of-scale-decentralization and computation. Management Sci., 11:213–235, 1964.

    Google Scholar 

  27. F. Margot, M. Queyranne, and Y. Wang. Decompositions, network flows and a precedence constrained single machine scheduling problem. Unpublished manuscript, December, 1996.

    Google Scholar 

  28. A. Munier and J. C. König. A heuristic for a scheduling problem with communication delays. Oper. Res., 45:145–147, 1997.

    MATH  Google Scholar 

  29. C. A. Phillips, C. Stein, and J. Wein. Minimizing average completion time in the presence of release dates. Math. Programming B, 1998. To appear.

    Google Scholar 

  30. C. N. Potts. An algorithm for the single machine sequencing problem with precedence constraints. Math. Programming Stud., 13:78–87, 1980.

    MATH  MathSciNet  Google Scholar 

  31. M. Queyranne. Structure of a simple scheduling polyhedron. Math. Programming, 58:263–285, 1993.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  33. R. Ravi, A. Agrawal, and P. Klein. Ordering problems approximated: single-processor scheduling and interval graph completion. In Proceedings of the 18th International Colloquium on Automata, Languages, and Processing, Lecture Notes in Computer Science 510, pages 751–762, 1991.

    MATH  MathSciNet  Google Scholar 

  34. A. S. Schulz and M. Skutella. Personal communication, 1997.

    Google Scholar 

  35. A.S. Schulz. Scheduling and Polytopes. PhD thesis, Technical University of Berlin, 1996.

    Google Scholar 

  36. D. B. Shraoys, é. Tardos, and K. I. Aardal. Approximation algorithms for facility location problems. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 265–274, 1997.

    Google Scholar 

  37. J. B. Sidney. Decomposition algorithms for single-machine sequencing with precedence and deferral costs. Oper. Res., pages 283–298, 1975.

    Google Scholar 

  38. J. F. Stollsteimer. The effect of technical change and output expansion on the optimum number, size and location of pear marketing facilities in a California pear producing region. PhD thesis, University of California at Berkeley, Berkeley, California, 1961.

    Google Scholar 

  39. J. F. Stollsteimer. A working model for plant numbers and locations. J. Farm Econom., 45:631–645, 1963.

    Article  Google Scholar 

  40. M. Sviridenko. Personal communication, July, 1997.

    Google Scholar 

  41. L. A. Wolsey. Mixed integer programming formulations for production planning and scheduling problems. Invited talk at the 12th International Symposium on Mathematical Programming, MIT, Cambridge, August, 1985.

    Google Scholar 

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Klaus Jansen José Rolim

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Shmoys, D.B. (1998). Using linear programming in the design and analysis of approximation algorithms: Two illustrative problems. In: Jansen, K., Rolim, J. (eds) Approximation Algorithms for Combinatiorial Optimization. APPROX 1998. Lecture Notes in Computer Science, vol 1444. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0053960

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  • DOI: https://doi.org/10.1007/BFb0053960

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