Beyond Trees: The Weighted Center Problem on Gromov Hyperbolic Graphs

The Weighted Center} problem takes as its input a graph $G=(V,E)$ together with a profile $\pi$ such that every vertex $v$ is mapped to some nonnegative multiplicative weight $\pi(v)$. Its output must be some vertex $c$ minimizing $\max\{\pi(v)d_G(c,v) : v \in V\}$. The classic Center problem corresponds to the case where $\pi(v) =1$ for every vertex $v$. In the literature, various almost linear-time algorithms have been proposed for the Center problem on some well-structured classes of graphs. By contrast, similarly efficient algorithms for the Weighted Center problem have been scarce. We investigate how the Gromov hyperbolicity, alone or in combination with other metric and geometric properties on graphs, can be used in the design of exact and approximate almost linear-time algorithms for the Weighted Center problem. In particular, we derive almost optimal algorithms for the following well-studied classes of graphs: chordal graphs, distance-hereditary graphs (both in $\mathcal{O}(m)$ time), dually chordal graphs and chordal bipartite graphs (both in $\mathcal{O}(m\log{n})$ time).