On a Simple Sampling Lemma

Abstract

Random sampling is an efficient method to deal with constrained optimization problems in computational geometry. In a first step, one finds the optimal solution subject to a random subset of the constraints; in many cases, the expected number of constraints still violated by that solution is then significantly smaller than the overall number of constraints that remain. This phenomenon can be exploited in several ways, and typically results in simple and asymptotically fast algorithms.Virtually every analysis of random sampling in this context boils down to a simple identity (the sampling lemma) which holds in an amazingly general framework, yet has not explicitly been stated in the literature.We present the sampling lemma, along with some of its applications, and we outline further results in the specific scenario of LP-type problems.