The minimum \emph{Monitoring Edge-Geodetic Set} problem asks for a minimum vertex subset $M$ such that for every edge $e$ of the input graph $G$, there exist a two vertices $u,v \in M$, that monitor $e$, i.e. every shortest path between $u$ and $v$ in $G$ contains $e$. We prove that this problem cannot be approximated within an $Ω(\log n)$ ratio in polynomial time unless $\mathsf{NP} \subseteq \mathsf{ZTIME}(n^{O(\log \log n)})$. Additionally, we establish hardness and conditional running-time lower bounds for exact algorithms on $2$-degenerate, $3$-degenerate, and $1$-apex $2$-degenerate graphs. Moreover, we design an approximation algorithm for hereditary graph classes with efficiently computable, truly sublinear-size balanced separators. This achieves an $O(n^{\frac{1}{4}} \sqrt{\log n})$ approximation ratio for $K_h$-minor-free graphs ($h=O(1)$), bounded genus graphs, and $k$-apex graphs ($k=O(n^{\frac{1}{4}})$). For graphs with bounded treewidth, the algorithm yields an $O(\log^{3/2} n)$ approximation ratio.

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