Random sampling in geometric optimization: new insights and applications

Very often the 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. 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 heuristic used in the simplex method for linear programming in order to reduce a large problem to only few small ones. This follows from our analysis of a reduction scheme for general LP-type problems, which can be considered as a simplification of algorithms by Clarkson [6] as well as Adler and Shamir [1]. The simplified version needs less random resources and allows a Chernoff-type tall estimate. 1. I N T R O D U C T I O N Random sampling and randomized incremental construction have become well-established, by now even classical, design paradigms in the field of computational geometry [21]. Many algorithms following that paradigm have been simplified to a point where they can easily be taught in introductory CS courses, with almost no technical difficulties. *The first author acknowledges support from the Swiss Science Foundation (SNF), project No. 21-50647.97. Permission to make digital or hard copies of all or part of this work tbr personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the lull citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a lee. Computational Geometry 2000 Hong Kong China Copyright ACM 2000 1-58113-224-7/00/6. $5.00 This was not always the case; pioneering papers, notably the ones by Clarkson and Shor [4, 7], and by Guibas, Knuth and Sharir [16], still required considerable technical derivations. This changed only when Seidel popularized the backwards analysis paradigm for randomized algorithms [24]. Together with the abstract framework of configuration spaces, this technique allows to treat many different algorithms in a simple and unified way [9]. The goal of this paper is to popularize and prove results around a simple identity (the sampling lemma) which underlies randomized algorithms for many geometric optimization problems. By that we mean problems defined over a lowdimensional space, which usually means that they have few constraints or few variables when written as mathematical programs. As we show below, special cases of the identity, or inequalities implied by it, are used in many places, including the analysis of the general configuration space framework. To the knowledge auf the authors, the identity itself, however, has not been noticed explicitly. The Sampl ing L e m m a Let S be a set of size n and ¢ a function that maps any set R C S to some value ¢(R). 1 Define v (R) := #{s ~ s \ R I ¢(R u {s}) ~ ¢(R)), X(R) : = #{8 e R I ¢(R \ {s}) ¢ ¢(R)}. V(R) is the set of violators of R, while X(R) is the set of extreme elements in R. Obviously, s violates R ¢~ s is extreme in R U {s}. (1) For R a random sample of size r, i.e. a set R uniformly chosen from the set (s) of all r-element subsets of S, we define random variables V~ : R ~-~ IV(R)I and Xr : R ~+ IX(R)h and we consider the expected values vr := E(VT), x, := E(X~). LEMMA 1.1. (Sampling Lemma). For 0 <_ r < n, (r + 1)vr = x ~ + l ( n r). 1Here, the only purpose of ¢ is to partition 2 s into equivalence classes; later, the function-notation becomes clear.