On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets

Davide Bilò; Giordano Colli; Luca Forlizzi; Stefano Leucci

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arXiv · 2405.13875

On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets

Davide Bilò·Giordano Colli·Luca Forlizzi·Stefano Leucci

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Abstract

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 $\mathsf{P} = \mathsf{NP}$.

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Davide Bilò, Giordano Colli, Luca Forlizzi, Stefano Leucci. 2024-05-22. On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets. https://arxiv.org/abs/2405.13875

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