Improved approximation algorithms for geometric set cover

Given a collection S of subsets of some set U, and M subset of U, the set cover problem is to find the smallest subcollection C subset of S that covers M, that is, M subset of boolean OR(C), where boolean OR(C) denotes boolean OR(Y is an element of C) Y. We assume of course that S covers M. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually U = R-d. Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R subset of S and nondecreasing function f (.), there is a decomposition of the complement U\boolean OR(R) into an expected at most f (vertical bar R vertical bar) regions, each region of a particular simple form. Under this condition, a cover of size O(f(vertical bar C vertical bar)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f (c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in R-3, and for guarding an x-monotone polygonal chain.