Exact Algorithms and APX-Hardness Results for Geometric Set Cover

We study several geometric set cover problems in which the goal is to compute a minimum cover of a given set of points in Euclidean space by a family of geometric objects. We give a short proof that this problem is APX-hard when the objects are axis-aligned fat rectangles, even when each rectangle is an ǫ-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algorithm for 2-dimensional set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required.