A Subquadratic Time Algorithm for the Weighted $k$-Center Problem on Cactus Graphs

The weighted $k$-center problem in graphs is a classical facility location problem where we place $k$ centers on the graph, which minimize the maximum weighted distance of a vertex to its nearest center. We study this problem when the underlying graph is a cactus with $n$ vertices and present an $O(n \log^2 n)$ time algorithm for the same. This time complexity improves upon the $O(n^2)$ time algorithm by Ben-Moshe et al. [TCS 2007], which is the current state-of-the-art.