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

2025-07-01

AI Summary

This paper studies the Minimum Edge Geodetic Set (MEGSET) problem: given a graph (G), find a smallest vertex subset (M) such that every edge of (G) lies on some shortest path between two vertices in (M). We first establish, under the assumption (P eq NP), that MEGSET admits no polynomial-time approximation algorithm with ratio better than (Omega(log n)); further, we strengthen the known NP-hardness result from 2-apex graphs to the broader class of 1-apex graphs. Leveraging graph-theoretic insights and complexity analysis—and exploiting efficiently computable sublinear balanced separators—we design a novel approximation algorithm for hereditary graph classes. Our algorithm achieves approximation ratios of (O(n^{1/4}sqrt{log n})) for planar graphs, bounded-genus graphs, and (k)-apex graphs; and (O(log^{3/2} n)) for bounded-treewidth graphs—significantly improving upon the general bound of (O(sqrt{n log n})).

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 = p. To the best of our knowledge, this is the first non-constant inapproximability result known for this problem. We also strengthen the known p-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 p-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 extsc{Set Cover} problem.

Problem

• Studies approximability of Monitoring Edge Geodetic Set problem.
• Proves non-constant inapproximability unless P equals NP.
• Designs algorithms for hereditary graph classes with sublinear separators.

Innovation

• Proves Ω(log n) inapproximability for MEGSET
• Strengthens NP-hardness to 1-apex graphs
• Designs O(n^1/4 √log n) approximation for planar graphs