Violator Spaces: Structure and Algorithms

Details Violator Spaces: Structure and Algorithms Violator Spaces: Structure and Algorithms

2006-06-20

arxiv.org/abs/cs/0606087v3

Computer Science

Discrete Mathematics

28 pages, 5 figures, extended abstract was presented at ESA 2006; author spelling fixed

Scientific paper

10.1007/11841036_36

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).

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