For every weight assignment $\pi$ to the vertices in a graph $G$, the radius function $r_\pi$ maps every vertex of $G$ to its largest weighted distance to the other vertices. The center problem asks to find a center, i.e., a vertex of $G$ that minimizes $r_\pi$. We here study some local properties of radius functions in graphs, and their algorithmic implications; our work is inspired by the nice property that in Euclidean spaces every local minimum of every radius function $r_\pi$ is a center. We study a discrete analogue of this property for graphs, which we name $G^p$-unimodality: specifically, every vertex that minimizes the radius function in its ball of radius $p$ must be a central vertex. While it has long been known since Dragan (1989) that graphs with $G$-unimodal radius functions $r_\pi$ are exactly the Helly graphs, the class of graphs with $G^2$-unimodal radius functions has not been studied insofar. We prove the latter class to be much larger than the Helly graphs, since it also comprises (weakly) bridged graphs, graphs with convex balls, and bipartite Helly graphs. Recently, using the $G$-unimodality of radius functions $r_\pi$, a randomized $\widetilde{\mathcal{O}}(\sqrt{n}m)$-time local search algorithm for the center problem on Helly graphs was proposed by Ducoffe (2023). Assuming the Hitting Set Conjecture (Abboud et al., 2016), we prove that a similar result for the class of graphs with $G^2$-unimodal radius functions is unlikely. However, we design local search algorithms (randomized or deterministic) for the center problem on many of its important subclasses.

翻译：对于图$G$中每个顶点上的权重分配$\pi$，半径函数$r_\pi$将$G$的每个顶点映射到其与其他顶点的最大加权距离。中心问题要求找到一个中心，即最小化$r_\pi$的$G$的顶点。本文研究图中半径函数的一些局部性质及其算法意义；我们的工作灵感来源于欧氏空间中每个半径函数$r_\pi$的每个局部最小值都是中心这一优美性质。我们研究了该性质在图上的离散模拟，并将其命名为$G^p$-单峰性：具体而言，每个在其半径为$p$的球内最小化半径函数的顶点必须是中心顶点。尽管自Dragan（1989）以来已知具有$G$-单峰半径函数$r_\pi$的图正是Helly图，但具有$G^2$-单峰半径函数的图类迄今尚未被研究。我们证明后一类图远大于Helly图，因为它还包含（弱）桥接图、具有凸球的图以及二分Helly图。最近，Ducoffe（2023）利用半径函数$r_\pi$的$G$-单峰性，为Helly图上的中心问题提出了一种随机$\widetilde{\mathcal{O}}(\sqrt{n}m)$时间局部搜索算法。基于命中集猜想（Abboud等人，2016），我们证明对于具有$G^2$-单峰半径函数的图类获得类似结果的可能性较低。然而，我们针对该图类中许多重要子类设计了中心问题的局部搜索算法（随机或确定性）。