## A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization (2000)

Venue: | Discr. Comput. Geometry |

Citations: | 22 - 3 self |

### BibTeX

@ARTICLE{Gärtner00asimple,

author = {Bernd Gärtner and Emo Welzl},

title = {A Simple Sampling Lemma: Analysis and Applications in Geometric Optimization},

journal = {Discr. Comput. Geometry},

year = {2000},

volume = {25},

pages = {569--590}

}

### OpenURL

### 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. Very often the analysis of random sampling in this context boils down to a simple identity (the sampling lemma) which holds in a general framework, yet has not been stated explicitly in the literature. In the more restricted but still general setting of LP-type problems, we prove tail estimates for the sampling lemma, giving Chernoff-type bounds for the number of constraints violated by the solution of a random subset. As an application, we provide the first theoretical analysis of multiple pricing, a heu...