Intersecting disks using two congruent disks

We consider the following Euclidean 2-center problem. Given n disks in the plane, find two smallest congruent disks such that every input disk intersects at least one of the two congruent disks. We present a deterministic algorithm for the problem that returns an optimal pair of congruent disks in $O(n^2{\log }^{3}n/\log \log n)$ time. We also present a randomized algorithm with $O(n^2{\log }^{2}n/\log \log n)$ expected time. These results improve upon the previously best deterministic and randomized algorithms, making a step closer to the optimal algorithms for the problem. We show that the same algorithms also work for two variants of the problem, the 2-piercing problem and the restricted 2-cover problem on disks. We also consider the 2-center problem and its two variants on n convex polygons, each with $O\left(1\right)$ vertices in the plane and present efficient algorithms for them.