Additive approximation algorithm for geodesic centers in $δ$-hyperbolic graphs
Dibyayan Chakraborty, Yann Vaxès
TL;DR
This work studies the k-Geodesic Center problem on δ-hyperbolic graphs, seeking a set of $k$ isometric paths minimizing the maximum distance to any vertex. The authors introduce a coarse shallow-pairing property and reduce the problem to a rooted variant, delivering an additive $O(δ)$-approximation via a two-stage approach and a careful transformation from rooted to non-rooted solutions. They prove the method runs in $O(mn^2\log n)$ time and yields a $(6τ(G)+1)$-additive bound, with a exact polynomial-time solution on trees ($δ=0$). An NP-hardness result on partial grids is established by adapting a known reduction, highlighting limits in general graph classes. The paper also contributes the coarse pairing framework, which may be of independent interest for other tree-like graph problems and potential extensions to planar settings.
Abstract
For an integer $k\geq 1$, 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{$δ$-hyperbolicity} measures how treelike a graph is from a metric point of view. Our main contribution in this paper is to provide an additive $O(δ)$-approximation algorithm for \textsc{$k$-Geodesic Center} on $δ$-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 $δ$-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 $k\geq 1$, $k$-\textsc{Geodesic Center} is NP-hard even on partial grids.