Maximizing the Area of Overlap of Two Unions of Disks under Rigid Motion

Abstract

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a trans- lation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 − � )- approximation algorithms for translations and for rigid motions, which run in O((nm/� 2 ) log(m/� )) and O((n 2 m 2 /� 3 ) log m)) time, respectively. For rigid motions, we can also compute a (1−� )-approximation in O((m 2 n 4/3 ∆ 1/3 /� 3 ) log n log m) time, where ∆ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1−� )-approximation al- gorithm for rigid motions that runs in O((m 2 /� 4 ) log(m/� ) log 2 m) time. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.