Offset polygon and annulus placement problems

The δ-annulus of a polygon P is the closed region containing all points in the plane at distance at most δ from the boundary of P. An inner (resp., outer) δ-offset polygon is the polygon defined by the inner (resp., outer) boundary of its δ-annulus. In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of δ, to cover as many points from S as possible by the δ-offset (or by the δ-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of δ such that there is a rigid transformation of the δ-offset (or the δ-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering k≤|S| points. These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases.