Violator Spaces: Structure and Algorithms

TL;DR

This paper introduces violator spaces, a new framework generalizing LP-type problems, and demonstrates their utility by adapting Clarkson's algorithms, leading to faster solutions for certain optimization problems.

Contribution

The paper defines violator spaces as a simpler, more natural generalization of LP-type problems and shows how existing algorithms extend to this new framework.

Findings

Violator spaces generalize LP-type problems.

Clarkson's algorithms are applicable to violator spaces.

Faster algorithms for P-matrix generalized linear complementarity problems.

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).