Violator spaces: Structure and algorithms

  • Authors:

  • B. Gärtner;J. Matoušek;L. Rüst;P. Škovroň

  • Affiliations:

  • Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland;Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic and Institute of Theoretical Computer Science, Charles University, 118 00 Prah ...;Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland;Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

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).