Almost optimal set covers in finite VC-dimension

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous Θ(log⋎X⋎) 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 findO(c)-sized covers. The first author was supported in part by NSF Grant CCR-90-02352 and Ecole Normale Supérieure. The second author's research was supported by the NSF and DARPA under Grant CCR-8908092, and by the NSF under Grants IRI-9116843 and CCR-9300079.