On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets

TL;DR

This work studies the minimum MEG-set problem, where a monitoring edge-geodetic set must witness every edge via shortest-path containment between some pair of its vertices. It establishes a tight inapproximability bound: no polynomial-time $(c \log n)$-approximation exists for any constant $c<\frac{1}{2}$ unless $\mathsf{P}=\mathsf{NP}$, via a Set Cover reduction that encodes set covers as MEG-sets in a constructed graph. The reduction uses multiple copies of a bipartite incidence gadget and leaf attachments so that minimal MEG-sets correspond to selecting a collection of sets covering the universe, thereby transferring hardness from Set Cover to MEG-sets. This result clarifies the limits of efficiently approximating edge-geodetic monitoring problems under standard complexity assumptions, highlighting a barrier at $c=\tfrac{1}{2}$ for logarithmic approximations.

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

Figures (1)

• Figure 1: The graph $G$ obtained by applying our reduction with $k=2$ to the Set Cover instance $\mathcal{I} = \langle X, \mathcal{S}\rangle$ with $\eta=5$, $h=4$, $S_1 = \{x_1, x_2, x_3\}$, $S_2 = \{x_2, x_3\}$, $S_3 = \{x_2, x_4, x_5\}$, and $S_4 = \{x_3, x_5\}$. To reduce clutter, the edges of the clique induced by the vertices $y_i$ (in the gray area) are not shown.

Theorems & Definitions (15)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof