Algorithms and complexity for monitoring edge-geodetic sets in graphs

TL;DR

This paper studies the computational problem MEG-set, where a monitoring edge-geodetic set (MEG-set) of a graph $G$ is a vertex subset that ensures every edge $e$ lies on all shortest paths between some pair of probes from the subset, with $\mathrm{meg}(G)$ the minimum size. It delivers a balanced mix of hardness and algorithmic results: strong NP-hardness and ETH-based lower bounds, APX-hardness, and no PTAS for restricted graph classes, alongside positive results including a polynomial-time algorithm for interval graphs, FPT algorithms parameterized by clique-width plus diameter and by treewidth on chordal graphs, and a $(\ln m-\ln\ln m+0.78)(OPT-1)$-approximation plus a $\sqrt{n\ln m}$-approximation, where $m$ and $n$ are the numbers of edges and vertices and $OPT$ is the optimum MEG-set size. The paper leverages MSO-based fixed-parameter techniques (Courcelle's theorem) for CW+diameter, a detailed DP over nice tree decompositions for chordal graphs, and a Set Cover reduction for approximation, complemented by tight hardness reductions from Vertex Cover. The results advance understanding of MEG-set complexity, offering efficient algorithms for specific graph classes and robust approximation and parameterized schemes, while identifying open questions on planar and chordal instances and on fundamental parameterizations.

Abstract

A monitoring edge-geodetic set of a graph is a subset $M$ of its vertices such that for every edge $e$ in the graph, deleting $e$ increases the distance between at least one pair of vertices in $M$. We study the following computational problem \textsc{MEG-set}: given a graph $G$ and an integer $k$, decide whether $G$ has a monitoring edge geodetic set of size at most $k$. We prove that the problem is NP-hard even for 2-apex 3-degenerate graphs, improving a result by Haslegrave (Discrete Applied Mathematics 2023). Additionally, we prove that the problem cannot be solved in subexponential-time, assuming the Exponential-Time Hypothesis, even for 3-degenerate graphs. Further, we prove that the optimization version of the problem is APX-hard, even for 4-degenerate graphs. Complementing these hardness results, we prove that the problem admits a polynomial-time algorithm for interval graphs, a fixed-parameter tractable algorithm for general graphs with clique-width plus diameter as the parameter, and a fixed-parameter tractable algorithm for chordal graphs with treewidth as the parameter. We also provide an approximation algorithm with factor $\ln m\cdot OPT$ and $\sqrt{n\ln m}$ for the optimization version of the problem, where $m$ is the number of edges, $n$ the number of vertices, and $OPT$ is the size of a minimum monitoring edge-geodetic set of the input graph.

Figures (1)

• Figure 1: Construction of $\hat{G}$ from $G = P_3$ as explained in the reduction.

Theorems & Definitions (36)

• Lemma 2.1: foucaud2024bounds
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 3.1
• proof
• Theorem 3.2