On geometric optimization with few violated constraints

Abstract

We investigate the problem of finding the best solution satisfying all but k of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl-the so-called LP-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance for the problem of computing the smallest circle enclosing all but k of the given n points in the plane, we obtain an O(n log n+k3 nε) algorithm; this improves previous results for k small compared with n but moderately growing. We also establish some results concerning general properties of LP-type problems. © 1995 Springer-Verlag New York Inc.