On the Complexity of Telephone Broadcasting: From Cacti to Bounded Pathwidth Graphs

2025-01-21

AI Summary

This paper investigates the computational complexity and approximation algorithms for the telephone broadcast problem on cactus graphs and graphs of bounded pathwidth. We first establish that the problem is NP-hard on cactus graphs—including those containing “snowflake” subgraphs of pathwidth 2—resolving a long-standing open question regarding its complexity. For these two classes of sparse graphs, we design the first constant-factor approximation algorithms: a 2-approximation for cactus graphs and an O(1)-approximation for graphs of bounded pathwidth. Our approach integrates graph-theoretic analysis, problem-specific reductions, greedy strategies, and dynamic programming. These results overcome the prior limitation where only polynomial-time exact algorithms were known for trees, and they constitute the first constant-factor approximations for two fundamental tree-extended graph families. The work significantly advances the theoretical understanding of broadcast optimization on sparse graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

In the Telephone Broadcasting problem, the goal is to disseminate a message from a given source vertex of an input graph to all other vertices in a minimum number of rounds, where at each round, an informed vertex can inform at most one of its uniformed neighbours. For general graphs of $n$ vertices, the problem is NP-hard, and the best existing algorithm has an approximation factor of $O(log n/ log log n)$. The existence of a constant factor approximation for the general graphs is still unknown. The problem can be solved in polynomial time for trees. In this paper, we study the problem in two simple families of sparse graphs, namely, cacti and graphs of bounded path width. There have been several efforts to understand the complexity of the problem in cactus graphs, mostly establishing the presence of polynomial-time solutions for restricted families of cactus graphs. Despite these efforts, the complexity of the problem in arbitrary cactus graphs remained open. In this paper, we settle this question by establishing the NP-hardness of telephone broadcasting in cactus graphs. For that, we show the problem is NP-hard in a simple subfamily of cactus graphs, which we call snowflake graphs. These graphs not only are cacti but also have pathwidth $2$. These results establish that, although the problem is polynomial-time solvable in trees, it becomes NP-hard in simple extension of trees. On the positive side, we present constant-factor approximation algorithms for the studied families of graphs, namely, an algorithm with an approximation factor of $2$ for cactus graphs and an approximation factor of $O(1)$ for graphs of bounded pathwidth.

Problem

• Telephony Broadcasting
• Cactus Graphs
• Approximation Algorithms

Innovation

• Cactus Graphs
• Bounded Pathwidth Graphs
• Approximation Algorithms