A fundamental question in computational geometry is for a dynamic collection
of geometric objects in Euclidean space, whether it is possible to maintain a
maximum independent set in polylogarithmic update time. Already, for a set of
intervals, it is known that no dynamic algorithm can maintain an exact maximum
independent set with sublinear update time. Therefore, the typical objective is
to explore the trade-off between update time and solution size. Substantial
efforts have been made in recent years to understand this question for various
families of geometric objects, such as intervals, hypercubes, hyperrectangles,
and fat objects.
We present the first fully dynamic approximation algorithm for disks of
arbitrary radii in the plane that maintains a constant-factor approximate
maximum independent set in polylogarithmic update time. First, we show that for
a fully dynamic set of n unit disks in the plane, a 12-approximate maximum
independent set can be maintained with worst-case update time O(log2n),
and optimal output-sensitive reporting. Moreover, this result generalizes to
fat objects of comparable sizes in any fixed dimension d, where the
approximation ratio depends on the dimension and the fatness parameter. Our
main result is that for a fully dynamic set of disks of arbitrary radii in the
plane, an O(1)-approximate maximum independent set can be maintained in
polylogarithmic expected amortized update time.Comment: Abstract is shortened to meet Arxiv's requirement on the number of
character