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 (m ≥ n) disjoint unit disks in the plane. We consider the problem of finding 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 hen e, we turn our attention to approximation algorithms. We give deterministic (1-€)-approximation algorithms for the maximum area of overlap under translation and rigid motion that run in O((nm= 2) log(m= )) and O((n2m2 = 3) log m)) time respectively. For the later, if is the diameter of set A, we get an (1-€)-approximation in O(m2 n4=3 1=3 log n log m3) time which is improvement when = o(n2). Under the condition that the maximum is at least a constant fraction of the area of B, we give a probabilistic (1-€)-approximation algorithm for rigid motions that runs in O((n2 = 5) log2(n= )) time and suceeds with probability at least 1 -e -c n -6.