On the (In)Approximability of the Monitoring Edge Geodetic Set Problem

TL;DR

This work studies the minimum MEG-Set problem, where an edge is monitored if it lies on all shortest paths between a pair of chosen vertices, and seeks small vertex sets that monitor all edges. It establishes strong inapproximability: no poly-time algorithm can achieve o(log n) approximation unless P=NP, and no (α log n) approximation for any constant α<1/2 unless NP⊆DTIME(n^{O(log log n)}); it also proves NP-hardness on graphs with apex 1 and provides approximation algorithms for planar, bounded-genus, k-apex (k=O(n^{1/4})), and bounded-treewidth graphs via sparse balanced vertex-separators, plus a generalized MEG-Set framework with edge lengths and vertex costs. The paper extends hardness results to apex-1 graphs, strengthens the approximability landscape beyond the trivial Set Cover reduction, and highlights an open question for planar graphs, while delivering practical approximation schemes for several sparse graph families. Overall, it sharpens the complexity picture of MEG-Set and offers algorithms suited to graphs with favorable separator structures.

Abstract

We study the minimum \emph{Monitoring Edge Geodetic Set} (\megset) problem introduced in [Foucaud et al., CALDAM'23]: given a graph $G$, we say that an edge is monitored by a pair $u,v$ of vertices if \emph{all} shortest paths between $u$ and $v$ traverse $e$; the goal of the problem consists in finding a subset $M$ of vertices of $G$ such that each edge of $G$ is monitored by at least one pair of vertices in $M$, and $|M|$ is minimized. In this paper, we prove that all polynomial-time approximation algorithms for the minimum \megset problem must have an approximation ratio of $Ω(\log n)$, unless \p = \np. To the best of our knowledge, this is the first non-constant inapproximability result known for this problem. We also strengthen the known \np-hardness of the problem on $2$-apex graphs by showing that the same result holds for $1$-apex graphs. This leaves open the problem of determining whether the problem remains \np-hard on planar (i.e., $0$-apex) graphs. On the positive side, we design an algorithm that computes good approximate solutions for hereditary graph classes that admit efficiently computable balanced separators of truly sublinear size. This immediately results in polynomial-time approximation algorithms achieving an approximation ratio of $O(n^{\frac{1}{4}} \sqrt{\log n})$ on planar graphs, graphs with bounded genus, and $k$-apex graphs with $k=O(n^{\frac{1}{4}})$. On graphs with bounded treewidth, we obtain an approximation ratio of $O(\log^{3/2} n)$ for any constant $\varepsilon > 0$. This compares favorably with the best-known approximation algorithm for general graphs, which achieves an approximation ratio of $O(\sqrt{n \log n})$ via a simple reduction to the \textsc{Set Cover} problem.

Figures (2)

• Figure 1: An example of the construction of the graph $H$ from the input graph $G = (\{a,b,c,d\},\{ \; \{a,b\}, \{b,c\}, \{c,d\}, \{d,a\} \; \})$, which is shown in black.
• Figure 2: 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 (8)

• Lemma 1: Colli2023Monitoring_edge-geodetic_sets_in_graphs
• Lemma 2: Colli2023FoucaudMMSST24
• Lemma 3
• Lemma 4
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 8