On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets

TL;DR

This paper proves that approximating the minimum Monitoring Edge-Geodetic Set within a factor better than half the logarithm of the number of vertices is NP-hard, highlighting the problem's computational difficulty.

Contribution

It establishes a tight inapproximability bound for the minimum MEG-set problem, showing no efficient approximation within a logarithmic factor unless P=NP.

Findings

No polynomial-time $(c \, \log n)$-approximation exists for $c<\frac{1}{2}$ unless P=NP.

The problem is computationally hard to approximate within a logarithmic factor.

Theoretical bounds on the inapproximability of the MEG-set problem.

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