On Hardness and Approximation of Broadcasting in Sparse Graphs

TL;DR

This work sharpens the understanding of Telephone Broadcasting on sparse graphs by proving NP-hardness for the simple $k$-cycle and $k$-path graph families, while simultaneously delivering PTASs for both and an exact polynomial-time algorithm for graphs with bounded bandwidth. The hardness results hinge on reductions from Restricted Numerical 3-Dimensional Matching (RN3DM), including a novel Even-Odd RN3DM variant, which encode scheduling constraints into simple graph topologies. The PTASs arise from translating broadcasting into integer covering problems (Prefix Covering for cycles and Double Prefix Covering for paths) and applying rounding techniques to obtain constant-support instances solvable by exhaustive search; bounded-bandwidth graphs are handled via a dynamic-programming approach over a sliding window. Together, these results delineate a clear boundary between tractable and intractable cases in sparse graph classes and offer practical approximation schemes for relevant network dissemination scenarios.

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.

Figures (7)

• Figure 1: A snowflake graph (cactus of pathwdith 2) aminian2025complexity
• Figure 2: Graph of distance 1 to path forest egami2025broadcasting
• Figure 3: A $k$-cycle graph (Theorem \ref{['Theorem:HardnessKCylce']})
• Figure 4: A $k$-path graph (Theorem \ref{['Theorem:HardnessKPath']})
• Figure 6: An instance $C=\{7,7,9,13\}$ of Even-Odd RN3DM reduces to the above instance of telephone broadcasting in $k$-cylce graphs (with $s$ as the originator and $br = 8$). The broadcast tree associated with with solution $\alpha = (4,6,2,8)$ and $\beta = (3,1,7,5)$ is highlighted.

Theorems & Definitions (53)

• Definition 2.1: Telephone Broadcasting hedetniemi1988broadsurvey
• Definition 2.2: $k$-cycle and $k$-path graphs
• Definition 2.3: Graph Bandwidth
• Definition 3.1: Restricted Numerical 3-Dimensional Matching (RN3DM) num-matching-hardness
• Example 1: Yes-instance of RN3DM
• Definition 3.2: Even-Odd Restricted Numerical 3-Dimensional Matching (Even-Odd RN3DM)
• Example 2: Yes-instance of Even-Odd RN3DM
• Lemma 3.3
• proof : Sketch
• proof