On Geometric Set Cover for Orthants.

We study SET COVER for orthants: Given a set of points in a d-dimensional Euclidean space and a set of orthants of the form (-infinity, p(1)] x ... x (-infinity, p(d)], select a minimum number of orthants so that every point is contained in at least one selected orthant. This problem draws its motivation from applications in multi-objective optimization problems. While for d = 2 the problem can be solved in polynomial time, for d > 2 no algorithm is known that avoids the enumeration of all size-k subsets of the input to test whether there is a set cover of size k. Our contribution is a precise understanding of the complexity of this problem in any dimension d >= 3, when k is considered a parameter: For d = 3, we give an algorithm with runtime n(O(root k)), thus avoiding exhaustive enumeration. For d = 3, we prove a tight lower bound of n(Omega(root k)) (assuming ETH). For d >= 4, we prove a tight lower bound of n(Omega(k)) (assuming ETH). Here n is the size of the set of points plus the size of the set of orthants. The first statement comes as a corollary of a more general result: an algorithm for SET COVER for half-spaces in dimension 3. In particular, we show that given a set of points U in R-3, a set of half-spaces D in R-3, and an integer k, one can decide whether U can be covered by the union of at most k half-spaces from D in time vertical bar D vertical bar(O(root k)) . vertical bar U vertical bar(O(1)). We also study approximation for SET COVER for orthants. While in dimension 3 a PTAS can be inferred from existing results, we show that in dimension 4 and larger, there is no 1.05-approximation algorithm with runtime f(k) . n(o(k)) for any computable f, where k is the optimum.