We consider the Euclidean 2-center problem for a set of $n$ disks in the plane: find two smallest congruent disks such that every disk in the set 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 the previously best deterministic and randomized algorithms, making another step closer to the optimal algorithms for the problem. We show that the same algorithms also work for the 2-piercing problem and the restricted 2-covering problem on disks.

https://link.springer.com/chapter/10.1007/978-3-030-79987-8_28