A vertex in a graph is called central if it minimizes its maximum distance to the other vertices. The radius of a graph $G$ is the largest distance between a central vertex and the other vertices, and it is denoted by $rad(G)$. In the center problem, we are asked to find a central vertex. We study the fine-grained complexity of the center problem on graphs with small Gromov hyperbolicity. Roughly, the Gromov hyperbolicity of a graph represents how close, locally, it is to a tree, from a metric point of view. It has applications in the design of approximation algorithms. In particular, there is a linear-time algorithm that for every $δ$-hyperbolic graph $G$ outputs some vertex at distance at most $rad(G) + 5δ$ to the other vertices [Chepoi et al, SoCG'08]. However, a linear-time algorithm for computing a central vertex is known only for $0$-hyperbolic graphs, whereas its existence was ruled out for $2$-hyperbolic graphs under the Hitting Set Conjecture of [Abboud et al, SODA'16]. Our main contribution in the paper is a linear-time algorithm for computing a central vertex in the class of $\frac 1 2$-hyperbolic graphs. Furthermore, we rule out the existence of such an algorithm for $1$-hyperbolic graphs, under the Hitting Set Conjecture, thus completely settling all the cases left open.

翻译：图中的一个顶点若使其到其他顶点的最大距离最小化，则称为中心顶点。图$G$的半径是中心顶点到其他顶点的最大距离，记作$rad(G)$。中心问题要求我们找出一个中心顶点。我们研究了小Gromov双曲性图上中心问题的细粒度复杂度。粗略地说，图的Gromov双曲性从度量角度表征了其在局部上与树的接近程度。该性质在近似算法设计中具有应用价值。特别地，存在一个线性时间算法，对于每个$δ$-双曲图$G$，输出某个到其他顶点距离不超过$rad(G) + 5δ$的顶点[Chepoi等人，SoCG'08]。然而，仅在$0$-双曲图类中存在计算中心顶点的线性时间算法，而由于[Abboud等人，SODA'16]中的Hitting Set猜想，$2$-双曲图类中此类算法的存在性被排除。本文的主要贡献是为$\frac{1}{2}$-双曲图类提出了一个计算中心顶点的线性时间算法。此外，我们在Hitting Set猜想下排除了$1$-双曲图类中存在此类算法的可能性，从而完全解决了所有遗留情况。