Given an undirected connected graph $G = (V(G), E(G))$ on $n$ vertices, the minimum Monitoring Edge-Geodetic Set (MEG-set) problem asks to find a subset $M \subseteq V(G)$ of minimum cardinality such that, for every edge $e \in E(G)$, there exist $x,y \in M$ for which all shortest paths between $x$ and $y$ in $G$ traverse $e$. We show that, for any constant $c < \frac{1}{2}$, no polynomial-time $(c \log n)$-approximation algorithm for the minimum MEG-set problem exists, unless $\mathsf{P} = \mathsf{NP}$.

翻译：给定一个具有 $n$ 个顶点的无向连通图 $G = (V(G), E(G))$，最小监控边测地集问题要求找到一个最小基数的子集 $M \subseteq V(G)$，使得对于每条边 $e \in E(G)$，都存在 $x,y \in M$，使得 $G$ 中所有 $x$ 与 $y$ 之间的最短路径都经过 $e$。我们证明，对于任意常数 $c < \frac{1}{2}$，除非 $\mathsf{P} = \mathsf{NP}$，否则不存在针对最小监控边测地集问题的多项式时间 $(c \log n)$-近似算法。