Violator Spaces: Structure and Algorithms

  • Bernd Gärtner
  • Jiří Matoušek
  • Leo Rüst
  • Petr Škovroň
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson’s randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).


Linear Complementarity Problem Helly Theorem Generalize Linear Programming Tukey Depth Generalize Linear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bernd Gärtner
    • 1
  • Jiří Matoušek
    • 2
  • Leo Rüst
    • 1
  • Petr Škovroň
    • 2
  1. 1.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  2. 2.Department of Applied Mathematics and Institute of Theoretical Computer ScienceCharles UniversityPraha 1Czech Republic

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