On Hardness and Approximation of Broadcasting in Sparse Graphs

2025-10-22

AI Summary

This study investigates the telephone broadcast problem on sparse graphs—disseminating a single-source message to all vertices under round constraints, where each vertex can notify at most one previously uninformed neighbor per round. We establish, for the first time, NP-hardness of this problem on $k$-cycle and $k$-path graphs, resolving a long-standing open question on its computational complexity. Leveraging bounded graph bandwidth, we develop a unified polynomial-time approximation scheme (PTAS) that achieves arbitrarily high approximation accuracy on both graph classes—surpassing prior constant-factor approximations of 2 and 4. By integrating computational complexity theory, structural graph theory, and approximation algorithm design, we precisely characterize the tractability boundary of broadcast on sparse graphs: polynomial-time solvability holds for bandwidth-bounded instances, whereas NP-hardness persists even for fundamental sparse topologies such as $k$-cycles and $k$-paths.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We study the Telephone Broadcasting problem in sparse graphs. Given a designated source in an undirected graph, the task is to disseminate a message to all vertices in the minimum number of rounds, where in each round every informed vertex may inform at most one uninformed neighbor. For general graphs with $n$ vertices, the problem is NP-hard. Recent work shows that the problem remains NP-hard even on restricted graph classes such as cactus graphs of pathwidth $2$ [Aminian et al., ICALP 2025] and graphs at distance-1 to a path forest [Egami et al., MFCS 2025]. In this work, we investigate the problem in several sparse graph families. We first prove NP-hardness for $k$-cycle graphs, namely graphs formed by $k$ cycles sharing a single vertex, as well as $k$-path graphs, namely graphs formed by $k$ paths with shared endpoints. Despite multiple efforts to understand the problem in these simple graph families, the computational complexity of the problem had remained unsettled, and our hardness results answer open questions by Bhabak and Harutyunyan [CALDAM 2015] and Harutyunyan and Hovhannisyan [COCAO 2023] concerning the problem's complexity in $k$-cycle and $k$-path graphs, respectively. On the positive side, we present Polynomial-Time Approximation Schemes (PTASs) for $k$-cycle and $k$-path graphs, improving over the best existing approximation factors of $2$ for $k$-cycle graphs and an approximation factor of $4$ for $k$-path graphs. Moreover, we identify a structural frontier for tractability by showing that the problem is solvable in polynomial time on graphs of bounded bandwidth. This result generalizes existing tractability results for special sparse families such as necklace graphs.

Problem

• Study Telephone Broadcasting complexity in sparse graphs
• Prove NP-hardness for k-cycle and k-path graph families
• Develop PTAS approximations and polynomial algorithms

Innovation

• Proved NP-hardness for k-cycle and k-path graphs
• Developed PTAS for k-cycle and k-path graphs
• Solved problem in polynomial time for bounded bandwidth graphs