Chapter 23 Approximation Using Shell Sets

x " And so ended Svejk's Budejovice anabasis. It is certain that if Svejk had been granted liberty of movement he would have got to Budejovice on his own. However much the authorities may boast that it was they who brought Svejk to his place of duty, this is nothing but a mistake. With Svejk energy and irresistible desire to fight, the authorities action was like throwing a spanner into the works. " – The good soldier Svejk, Jaroslav Hasek. In this chapter we define a concept that is slightly weaker than coreset, and who how a reweight-ing algorithm can approximate it quickly. 23.1 Covering problems, expansion and shell sets Consider a set P of n points in IR d , that we are interested in covering by the best shape in a family of shapes F. For example, F might be the set of all balls in IR d , and we are looking for the minimum enclosing ball of P. A ε-coreset S ⊆ P would guarantee that any ball that covers S will cover the whole point set if we expand it by (1 + ε). However, sometimes, computing the coreset is computationally expensive, the coreset does not exist at all, or its size is prohibitively large. It is still natural to look for a small subset V of the points, such that finding the optimal solution for V generates (after appropriate expansion) an approximate solution to the original problem. So, let assume we are given a " slow " procedure algSlow that can solve the covering/clustering problem we are interested in. The basic idea would be to run the algorithm on a very small subset and use this solution for the whole point set. To this end, we assume the following: (A) There is a cost function f : F → IR that scores possible solutions, and we are looking for the cheapest range r ∈ F (according to f) that covers the input P ⊆ IR d .