Additive approximation algorithm for geodesic centers in $\hat{I}´$-hyperbolic graphs

Abstract

For an integer $k\ge 1$, the objective of \textsc{$k$-Geodesic Center} is to find a set $\mathcal{C}$ of $k$ isometric paths such that the maximum distance between any vertex $v$ and $\mathcal{C}$ is minimised. Introduced by Gromov, \emph{$\hat{I}´$-hyperbolicity} measures how treelike a graph is from a metric point of view. Our main contribution in this paper is to provide an additive $O(\hat{I}´)$-approximation algorithm for \textsc{$k$-Geodesic Center} on $\hat{I}´$-hyperbolic graphs. On the way, we define a coarse version of the pairing property introduced by Gerstel \& Zaks (Networks, 1994) and show it holds for $\hat{I}´$-hyperbolic graphs. This result allows to reduce the \textsc{$k$-Geodesic Center} problem to its rooted counterpart, a main idea behind our algorithm. We also adapt a technique of Dragan \& Leitert, (TCS, 2017) to show that for every $k\ge 1$, $k$-\textsc{Geodesic Center} is NP-hard even on partial grids.