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Improving Linear Programming Approaches for the Steiner Tree Problem

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Experimental and Efficient Algorithms (WEA 2003)

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Abstract

We present two theoretically interesting and empirically successful techniques for improving the linear programming approaches, namely graph transformation and local cuts, in the context of the Steiner problem. We show the impact of these techniques on the solution of the largest benchmark instances ever solved.

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References

  1. E. Althaus, T. Polzin, and S. Vahdati Daneshmand. Improving linear programming approaches for the steiner tree problem. Research Report MPI-I-2003-1-004, Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany, February 2003.

    Google Scholar 

  2. Y. P. Aneja. An integer linear programming approach to the Steiner problem in graphs. Networks, 10:167–178, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Applegate, R. Bixby, V. Chvátal, and W. Cook. Finding cuts in the TSP (A preliminary report). Technical report, Center for Discrete Mathematics and Theoretical Computer Science, Rutgers University, Piscataway, NJ, 1995.

    Google Scholar 

  4. D. Applegate, R. Bixby, V. Chvátal, and W. Cook. TSP cuts which do not conform to the template paradigm. In Michael Junger and Denis Naddef, editors, Computational Combinatorial Optimization, volume 2241 of Lecture Notes in Computer Science. Springer, 2001.

    Chapter  Google Scholar 

  5. E. Balas and M. Padberg. On the set-covering problem: II. An algorithm for set partitioning. Operations Research, 23:74–90, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  6. X. Cheng and D.-Z. Du, editors. Steiner Trees in Industry, volume 11 of Combinatorial Optimization. Kluwer Academic Publishers, Dordrecht, 2001.

    Google Scholar 

  7. S. Chopra and M. R. Rao. The Steiner tree problem I: Formulations, compositions and extension of facets. Mathematical Programming, pages 209–229, 1994.

    Google Scholar 

  8. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, 1990.

    Google Scholar 

  9. C. Gentile, U.-U. Haus, M. Köppe, G. Rinaldi, and R. Weismantel. A primal approach to the stable set problem. In R. Möhring and R. Raman, editors, Algorithms — ESA 2002, volume 2461 of Lecture Notes in Computer Science, pages 525–537, Rom, Italy, 2002. Springer.

    Chapter  Google Scholar 

  10. U.-U. Haus, M. Köppe, and R. Weismantel. The integral basis method for integer programming. Mathematical Methods of Operations Research, 53(3):353–361, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. K. Hwang, D. S. Richards, and P. Winter. The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. North-Holland, Amsterdam, 1992.

    MATH  Google Scholar 

  12. R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum Press, New York, 1972.

    Google Scholar 

  13. T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207–232, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Naddef and S. Thienel. Efficient separation routines for the symmetric traveling salesman problem i: general tools and comb separation. Mathematical Programming, 92(2):237–255, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Naddef and S. Thienel. Efficient separation routines for the symmetric traveling salesman problem ii: separating multi handle inequalities. Mathematical Programming, 92(2):257–283, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Polzin and S. Vahdati Daneshmand. A comparison of Steiner tree relaxations. Discrete Applied Mathematics, 112:241–261, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Polzin and S. Vahdati Daneshmand. Improved algorithms for the Steiner problem in networks. Discrete Applied Mathematics, 112:263–300, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Polzin and S. Vahdati Daneshmand. Partitioning techniques for the Steiner problem. Research Report MPI-I-2001-1-006, Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany, 2001.

    Google Scholar 

  19. T. Polzin and S. Vahdati Daneshmand. Extending reduction techniques for the steiner tree problem. In R. Möhring and R. Raman, editors, Algorithms — ESA 2002, volume 2461 of Lecture Notes in Computer Science, pages 795–807, Rom, Italy, 2002. Springer

    Chapter  Google Scholar 

  20. T. Polzin and S. Vahdati Daneshmand. On Steiner trees and minimum spanning trees in hypergraphs. Operations Research Letters, 31, 2003.

    Google Scholar 

  21. G. Reinelt. TSPLIB — a traveling salesman problem library. ORSA Journal on Computing, 3:376–384, 1991.

    MATH  Google Scholar 

  22. SteinLib. http://elib.zib.de/steinlib, 1997. T. Koch, A. Martin, and S. Voß.

  23. D. M. Warme, P. Winter, and M. Zachariasen. Exact algorithms for plane Steiner tree problems: A computational study. In D-Z. Du, J. M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 81–116. Kluwer Academic Publishers, 2000.

    Google Scholar 

  24. D. M. Warme, P. Winter, and M. Zachariasen. GeoSteiner 3.1. http://www.diku.dk/geosteiner/, 2001.

  25. R. T. Wong. A dual ascent approach for Steiner tree problems on a directed graph. Mathematical Programming, 28:271–287, 1984.

    Article  MATH  MathSciNet  Google Scholar 

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Althaus, E., Polzin, T., Daneshmand, S.V. (2003). Improving Linear Programming Approaches for the Steiner Tree Problem. In: Jansen, K., Margraf, M., Mastrolilli, M., Rolim, J.D.P. (eds) Experimental and Efficient Algorithms. WEA 2003. Lecture Notes in Computer Science, vol 2647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44867-5_1

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  • DOI: https://doi.org/10.1007/3-540-44867-5_1

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  • Print ISBN: 978-3-540-40205-3

  • Online ISBN: 978-3-540-44867-9

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