Almost optimal set covers in finite VC-dimension

Abstract

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimension d such that the size of our cover is at most a factor of O(d log(dc)) from the optimal size, c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives an O(log c) approximation factor. This improves the previous Θ(log{curly logical or}X{curly logical or}) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly find O(c)-sized covers. © 1995 Springer-Verlag New York Inc.