Algorithms and complexity for monitoring edge-geodetic sets in graphs

2024-09-27

arXiv.org

AI Summary

This paper investigates the Monitoring Edge Geodetic Set (MEGS) problem: given a graph $G$ and an integer $k$, determine whether there exists a vertex subset $M subseteq V(G)$ of size at most $k$ such that, for every edge $e in E(G)$, the deletion of $e$ strictly increases the shortest-path distance between some pair of vertices in $M$. We establish that MEGS is NP-hard, not solvable in subexponential time under the Exponential Time Hypothesis (ETH), and APX-hard. We design the first $O(n+m)$ exact algorithm for interval graphs. For general graphs and chordal graphs, we develop fixed-parameter tractable (FPT) algorithms parameterized by clique-width plus diameter and by treewidth, respectively. Additionally, we present a bi-criteria approximation algorithm achieving guarantees of $ln m cdot mathrm{OPT}$ and $sqrt{n ln m}$. Our results provide a tight complexity characterization and breakthroughs in efficient solvability across multiple structured graph classes.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 extsc{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 $ln mcdot OPT$ and $sqrt{nln m}$ 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.

Problem

• Determining NP-hardness of monitoring edge-geodetic sets in 2-apex 3-degenerate graphs
• Proving subexponential-time unsolvability under ETH for 3-degenerate graphs
• Developing FPT algorithms for interval and chordal graphs with specific parameters

Innovation

• NP-hard proof for 2-apex 3-degenerate graphs
• FPT algorithms for clique-width and treewidth
• Approximation algorithm with ln(m) factor