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

Let A and B be two sets of n resp. m (m n) disjoint unit disks in the plane. We consider the problem of nding a rigid motion of A that maximizes the total area of its 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. First, we give a deterministic (1 )-approximation algorithm for the maximum area of overlap under rigid motion that runs in O((nm= ) logm)) time. If is the diameter of set A, we get an (1 )-approximation in O( n 1=3 logn logm 3 ) time. Under the condition that the maximum is at least a constant fraction of the area of A, we give a probabilistic (1 )approximation algorithm that runs in O((m= ) log(m= ) logm) time and succeeds with high probability. Our algorithms generalize to the case where A and B consist of possibly intersecting disks of di erent 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.