A branch-and-cut approach for minimum weight triangulation

  • Yoshiaki Kyoda
  • Keiko Imai
  • Fumihiko Takeuchi
  • Akira Tajima
Session 8A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)

Abstract

This paper considers the problem of computing a minimum weight triangulation of n points in the plane, which has been intensively studied in recent years in computational geometry. This paper investigates a branch-and-cut approach for minimum weight triangulations. The problem can be formulated as finding a minimum-weight maximal independent set of intersection graphs of edges. In combinatorial optimization, there are known many cuts for the independent set problem, and we further use a cut induced by geometric properties of triangulations. Combining this branch-and-cut approach with the β-skeleton method, the moderate-size problem could be solved efficiently in our computational experiments. Polyhedral characterizations of the proposed cut and applications of another old skeletal approach in mathematical programming as the independent set problem are also touched upon.

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References

  1. 1.
    D. Applegate, R. Bixby, V. Chvatal and B. Cook. Finding Cuts in the TSP (A preliminary report). Technical Reports 95-05, DIMACS, 1995.Google Scholar
  2. 2.
    M. Berkelaar. lp_solve. Available at ftp://ftp.es.ele.tue.nl/pub/lp_solve.Google Scholar
  3. 3.
    M. Bern and D. Eppstein. Mesh generation and optimal triangulation. In “Computing in Euclidean Geometry” (2nd edition), Lecture Notes Series on Computing, Vol.4, World Scientific, 1995, pp.47–123.Google Scholar
  4. 4.
    S.-W. Cheng, M. J. Golin, and J. C. F. Tsang. Expected case analysis of β-skeletons with applications to the construction of minimum-weight triangulations. Proc. 7th Canadian Conference of Computational Geometry, 1995, pp.279–284.Google Scholar
  5. 5.
    S.-W. Cheng, N. Katoh and M. Sugai. A study of the LMT-skeleton. Proc. 7th International Symposium on Algorithms and Computation, Lecture Notes in Computer Science, Vol.1178, pp.256–265.Google Scholar
  6. 6.
    S.-W. Cheng and Y.-F. Xu. Approaching the largest β-Skeleton within a minimum weight triangulation. Proc. 12th Annual ACM Symposium on Computational Geometry, 1996, pp.196–203.Google Scholar
  7. 7.
    M. T. Dickerson and M. H. Montague. A (usually?) connected subgraph of the minimum weight triangulation. Proc. 12th Annual ACM Symposium on Computational Geometry, 1996, pp.204–213.Google Scholar
  8. 8.
    H. Edelsbrunner and T. S. Tan. A quadratic time algorithm for the minmax length triangulation. SIAM J. Comput., 22:527–551, 1993.Google Scholar
  9. 9.
    J. M. Keil. Computing a subgraph of the minimum weight triangulation. Computational Geometry Theory and Applications, 4:13–26, 1994.Google Scholar
  10. 10.
    Y. Kyoda. A study of generating minimum weight triangulation within practical time. Master's Thesis, Department of Information Science, University of Tokyo, March 1996. Available at http://naomi.is.s.u-Tokyo.ac.jp/Google Scholar
  11. 11.
    C. Levcopoulos and D. Krznaric. Quasigreedy triangulations approximating the minimum weight triangulation. Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, 1996, pp.392–401.Google Scholar
  12. 12.
    T. Masada, H. Imai and K. Imai. Enumeration of regular triangulations. Proc. 12th Annual ACM Symposium on Computational Geometry, 1996, pp.224–233.Google Scholar
  13. 13.
    G. L. Nemhauser and L. E. Trotter. Vertex packings: structural properties and algorithms. Mathematical Programming, 8,2:232–248, 1975.Google Scholar
  14. 14.
    F. Takeuchi and H. Imai. Enumerating triangulations for products of two simplices and for arbitrary configurations of points. Proc. 3rd International Computing and Combinatorics Computing Conference, Lecture Notes in Computer Science, 1997.Google Scholar
  15. 15.
    C. A. Wang, F. Chin and Y.-F. Xu. A new subgraph of minimum weight triangulation. Proc. 7th International Symposium on Algorithms and Computation, Lecture Notes in Computer Science, Vol.1178, pp.266–274.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Yoshiaki Kyoda
    • 1
  • Keiko Imai
    • 2
  • Fumihiko Takeuchi
    • 1
  • Akira Tajima
    • 3
  1. 1.Department of Information ScienceUniversity of TokyoJapan
  2. 2.Department of Information and System EngineeringChuo UniversityJapan
  3. 3.IBM Tokyo Research LaboratoryJapan

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