Clarkson's Algorithm for Violator Spaces

Clarkson’s algorithm is a two-staged randomized algorithm for solving linear programs. This algorithm has been simplified and adapted to fit the framework of LP-type problems. In this framework we can tackle a number of non-linear problems such as computing the smallest enclosing ball of a set of points in R. In 2006, it has been shown that the algorithm in its original form works for violator spaces too, which are a proper generalization of LP-type problems. It was not clear, however, whether previous simplifications of the algorithm carry over to the new setting. In this paper we show the following theoretical results: (a) It is shown, for the first time, that Clarkson’s second stage can be simplified. (b) The previous simplifications of Clarkson’s first stage carry over to the violator space setting. (c) Furthermore, we show the equivalence of violator spaces and partitions of the hypercube by hypercubes.