Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)-approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c is a constant that depends only on k and ε. This suggest that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.

Given a set of planar shapes (intervals, squares or disks) of the same size whose centers are fixed, we give the first polynomial-time algorithms to minimize (approximately) the common radius of the shapes under the requirement of large overlap with certain translated copies. The so-called auto-correlation function of a set of shapes at a vector x is the area of overlap between the set with its translated copy by vector x. The minimization is done with respect to an objective O and a threshold value µ, i.e. the auto-correlation function must not fall below µ within O. This type of problem is motivated by an application in optical interferometry, and the previous algorithms were of a heuristic nature, without any runtime or quality guarantees. Apart from fixing the centers, we had to restrict the original problem even further, in order to obtain the polynomial-time bounds. We consider the uniform case where all shapes are of the same size, and therefore our algorithms are not directly relevant for the motivating application. Rather, our research highlights the fact that even the simplified variants of the problem are difficult. The precise complexity status of these variants remains to be determined. Let n be the number of shapes involved. In particular, our results include an O(n2 logn) algorithm for minimizing the radius of uniform intervals in R2, an O(n4 logn) algorithm for axis-aligned uniform squares in R2, and a (10 9 + ǫ)-approximation algorithm for the case of uniform disk in R2 that runs in O(n6k2 lognk), where k is some sufficiently large integer depending on ǫ. The latter algorithm can be extended to a PTAS for uniform circles in R 2. Furthermore, in the appendix, we give some intuition why the problem of minimizing non-uniform sizes is difficult even for intervals in R1.