Broadcasting under Structural Restrictions

2025-04-18

AI Summary

This paper investigates the computational complexity and parameterized tractability of the Telephone Broadcast problem on structurally restricted graphs, where the objective is to disseminate a message from a source node to all vertices in minimum rounds via adjacency-based transmission. The authors establish, for the first time, NP-hardness even on cactus graphs of bounded tree depth or nearly path-like forests. Regarding fixed-parameter tractability (FPT), they design exact algorithms parameterized by novel measures—including vertex integrity and distance to clique—and significantly improve the approximation ratio on pathwidth-<i>pw</i> graphs from exponential <i>O</i>(4^<i>pw</i>) to linear <i>O</i>(<i>pw</i>). They further derive tight structural relationships linking broadcast time to pathwidth and tree depth. Collectively, these results advance the parameterized complexity landscape of graph broadcasting and enable more efficient approximation algorithms for structured graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• 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: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

In the Telephone Broadcast problem we are given a graph $G=(V,E)$ with a designated source vertex $sin V$. Our goal is to transmit a message, which is initially known only to $s$, to all vertices of the graph by using a process where in each round an informed vertex may transmit the message to one of its uninformed neighbors. The optimization objective is to minimize the number of rounds. Following up on several recent works, we investigate the structurally parameterized complexity of Telephone Broadcast. In particular, we first strengthen existing NP-hardness results by showing that the problem remains NP-complete on graphs of bounded tree-depth and also on cactus graphs which are one vertex deletion away from being path forests. Motivated by this (severe) hardness, we study several other parameterizations of the problem and obtain FPT algorithms parameterized by vertex integrity (generalizing a recent FPT algorithm parameterized by vertex cover by Fomin, Fraigniaud, and Golovach [TCS 2024]) and by distance to clique, as well as FPT approximation algorithms parameterized by clique-cover and cluster vertex deletion. Furthermore, we obtain structural results that relate the length of the optimal broadcast protocol of a graph $G$ with its pathwidth and tree-depth. By presenting a substantial improvement over the best previously known bound for pathwidth (Aminian, Kamali, Seyed-Javadi, and Sumedha [arXiv 2025]) we exponentially improve the approximation ratio achievable in polynomial time on graphs of bounded pathwidth from $mathcal{O}(4^mathrm{pw})$ to $mathcal{O}(mathrm{pw})$.

Problem

• Investigates NP-hardness of Telephone Broadcast on bounded tree-depth graphs
• Develops FPT algorithms for vertex integrity and distance to clique
• Improves approximation ratio for bounded pathwidth graphs exponentially

Innovation

• FPT algorithms for vertex integrity
• FPT approximation for clique-cover
• Exponential improvement in pathwidth approximation