Rectilinear and polygonal p-piercing and p-center problems

We consider the p-piercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or near-linear algorithms for small values of p in cases where the given regions are either axis-parallel rectangles or convex c-oriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions). We also investigate the planar rectilinear (and polygonal) p-center problem, in which we are given a set S of n points in the plane, and wish to find p axis-parallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a linear-time solution for the rectilinear 3-center problem (by showing that this problem can be formulated as an LP-type problem and by exhibiting a relation to Helly numbers). We give O(n log n)-time solutions for 4-piercing of translates of a square, as well as for the rectilinear 4-center problem; this is worst-case optimal. We give O(n polylog n)-time solutions for 4- and 5-piercing of axis-parallel rectangles, for more general rectilinear 4-center problems, and for rectilinear 5-center problems. 2-pierceability of a set of n convex c-oriented polygons can be decided in time O(c²n log n), and the 2-center problem for a convex c-gon can be solved in O(c⁵n log n) time. The first solution is worst-case optimal when c is fixed.