Discrete & Computational Geometry

, Volume 12, Issue 3, pp 241–261 | Cite as

Helly-type theorems and Generalized Linear Programming

  • N. Amenta


Recent combinatorial algorithms for linear programming can also be applied to certain nonlinear problems. We call these Generalized Linear-Programming, or GLP, problems. We connect this class to a collection of results from combinatorial geometry called Helly-type theorems. We show that there is a Helly-type theorem about the constraint set of every GLP problem. Given a familyH of sets with a Helly-type theorem, we give a paradigm for finding whether the intersection ofH is empty, by formulating the question as a GLP problem. This leads to many applications, including linear expected time algorithms for finding line transversals and mini-max hyperplane fitting. Our applications include GLP problems with the surprising property that the constraints are nonconvex or even disconnected.


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  1. [A]
    N. Amenta. Finding a line transversal of axial objects in three dimensionsProceeding of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, 1992, pages 66–71.Google Scholar
  2. [AGPW]
    B. Aronov, J. E. Goodman, R. Pollack, and R. Wenger. There is no Hadwiger number for line transversals in higher dimensions. Unpublished manuscript, cited in [GPW], Theorem 2.9.Google Scholar
  3. [AD]
    D. Avis and M. Doskas. Algorithms for high dimensional stabbing problems,Discrete Applied Mathematics, vol. 27 (1990), pages 39–48.MATHMathSciNetCrossRefGoogle Scholar
  4. [AH]
    D. Avis and M. E. Houle. Computational aspects of Helly’s theorem and its relatives,Proceedings of the Third Canadian Conference on Computational Geometry, 1991, pages 11–14.Google Scholar
  5. [CM]
    B. Chazelle and J. Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension,Proceeding of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, pages 281–290.Google Scholar
  6. [C]
    K. L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small, Manuscript, 1990. An earlier version appeared inProceedings of the 29th Annual Symposium on Foundations of Computer Science, 1988, pages 452–455.Google Scholar
  7. [DGK]
    L. Danzer, B. Grünbaum, and V. Klee Helly’s theorem and its relatives,Proceedings of the Symposium on Pure Mathematics, vol. 7, 1963, pages. 101–180. American Mathematical Society, Providence, RI.Google Scholar
  8. [D1]
    J. Demmel. The componentwise distance to the nearest singular matrix,SIAM Journal of Matrix Analysis and Applications, vol. 13, no. 1 (1992), pages 10–19.MATHMathSciNetCrossRefGoogle Scholar
  9. [D2]
    M. Dyer. On a multidimensional search technique and its application to the Euclidean one-center problem,SIAM Journal on Computing vol. 15 (1986), pages 725–738.MATHMathSciNetCrossRefGoogle Scholar
  10. [D3]
    M. Dyer. A class of convex programs with applications to computational geometry,Proceedings of the 8th Annual Symposium on Computational Geometry, 1992, pages 9–15.Google Scholar
  11. [E]
    J. Eckhoff. Helly, Radon- and Carathody type theorems, inHandbook of Convex Geometry, P. M. Gruber and J. M. Willis, eds., Chapter 2.1. Elsevier Science, Amsterdam, 1993.Google Scholar
  12. [EW]
    P. Egyed and R. Wenger. Stabbing pairwise disjoint translates in linear time,Proceedings of the 5th Annual Symposium on Computational Geometry, 1989, pages 364–369.Google Scholar
  13. [GPW]
    J.E. Goodman, R. Pollack, and R. Wenger, Geometric transversal theory, inNew Trends in Discrete and Computational Geometry. Springer-Verlag, New York (to appear).Google Scholar
  14. [GM]
    B. Grünbaum and T. S. Motzkin. On components in some families of sets,Proceedings of the American Mathematical Society, vol. 12 (1961), pages 607–613.MATHMathSciNetCrossRefGoogle Scholar
  15. [H]
    A. J. Hoffman. Binding constraints and Helly numbers,Annals of the New York Academy of Sciences, vol. 319 (1979), pages 284–288.CrossRefGoogle Scholar
  16. [K]
    G. Kalai. A subexponential randomized simplex algorithm,24th Annual ACM Symposium on the Theory of Computation, 1992, pages 475–482.Google Scholar
  17. [MSW]
    J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming,Proceedings of the 8th Annual Symposium on Computational Geometry, 1992, pages 1–8.Google Scholar
  18. [M1]
    N. Megiddo. Linear programming in linear time when the dimension is fixed.SIAM Journal on Computing, vol. 12 (1983), pages 759–776.MATHMathSciNetCrossRefGoogle Scholar
  19. [M2]
    N. Megiddo. On the ball spanned by balls,Discrete and Computational Geometry, vol. 4 (1989), pages 605–610.MATHMathSciNetCrossRefGoogle Scholar
  20. [M3]
    N. Megiddo. Personal communication (1991).Google Scholar
  21. [PR]
    S. Poljak and J. Rohn. Radius of nonsingularity,Mathematics of Systems, Signals and Control (to appear).Google Scholar
  22. [R]
    J. Rohn. Linear interval equations,Linear Algebra and Its Applications, vol. 126 (1989), pages 39–78.MATHMathSciNetCrossRefGoogle Scholar
  23. [S]
    R. Seidel. Linear programming and convex hulls made easy,Proceedings of the 6th Annual Symposium on Computational Geometry, 1990, pages 211–215.Google Scholar
  24. [SW]
    M. Sharir and E. Welzl. A combinatorial bound for linear programming and related problems,Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, 1992, pages 569–579. Lecture Notes in Computer Science, vol. 577 Springer-Verlag, Berlin.Google Scholar
  25. [T1]
    S. Toledo. Extremal polygon containment problems,Proceedings of the 7th Annual Symposium on Computational Geometry, 1991, pages 176–185.Google Scholar
  26. [T2]
    H. Tverberg. Proof of Grünbaum’s conjecture on common transversals for translates,Discrete and Computational Geometry, vol. 4 (1989), pages 191–203.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • N. Amenta
    • 1
    • 2
  1. 1.Computer ScienceUniversity of CaliforniaBerkeleyUSA
  2. 2.The Geometry CenterMinneapolisUSA

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