Almost optimal set covers in finite VC-dimension
Hervé Brönnimann, Michael T. Goodrich · 1994
We give a deterministic polynomial time method for finding a set cover in a set system (X,ℜ) of VC-dimension d such that the size of our cover is at most a factor of O(dlog(dc)) from the optimal size, c. For constant VC-dimension set systems, which are common in computational geometry, our method gives an O(logc) approximation factor. This improves the previous Θ(log |X|) bound of the greedy method and beats recent complexity-theoretic lower bounds for set covers (which don't make any assumptions about VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those that arise in 3-d polytope approximation and 2-d disc covering, we can quickly find O(c)-sized covers.