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

TL;DR

The paper addresses the Telephone Broadcasting problem on sparse graphs, proving NP-hardness for the simple snowflake family (pathwidth 2) while providing a practical 2-approximation for cactus graphs via the Cactus Broadcaster algorithm. It then establishes a chain of reductions from $3,4$-SAT to snowflake graphs, yielding NP-hardness for this class, and finally proves a constant-factor approximation for graphs of bounded pathwidth by leveraging Elkin's ${\mathcal O}\big(\frac{\log n}{\log {\boldsymbol{br}^*}(G,s)}\big)$ framework, with a lower bound ${\boldsymbol{br}^*}(G,s) \ge \Omega\big(n^{4^{-(w+1)}}\big)$ for pathwidth $w$. The combination of hardness results and approximation techniques delineates the complexity boundary between trees (polynomial-time) and more general sparse graphs, while delivering actionable approximation algorithms for cactus graphs and graphs with bounded pathwidth. The results have implications for distributed broadcasting in sparse networks and highlight open questions about constant-factor approximations for general graphs and potential PTASes for restricted graph classes.

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 the minimum number of rounds, where at each round, an informed vertex can send the message to at most one of its uninformed neighbors. For general graphs of n vertices, the problem is NP-complete, 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. In this paper, we study the problem in two simple families of sparse graphs, namely, cacti and graphs of bounded pathwidth. 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. We settle this question by establishing the NP-completeness of telephone broadcasting in cactus graphs. For that, we show the problem is NP-complete 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, despite being polynomial-time solvable in trees, the problem becomes NP-complete in very simple extensions 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.

Figures (19)

• Figure 1: An example of a reduced caterpillar, with the special vertices $x$, $y$, and $z$ highlighted
• Figure 2: An example of a snowflake graph
• Figure 4: The broadcast scheme in the $k$-broadcasting model with $k=2$
• Figure 5: The corresponding classic broadcasting scheme
• Figure 7: Two possible types for broadcasting to connected components after deleting $s$ in single-br($G,s$) are shown. The green vertices form a single-neighbor component. The red and blue components are double-neighbor components.

Theorems & Definitions (57)

• Definition 2.1: hedetniemi1988broadsurvey
• Definition 2.2
• Definition 2.3
• Definition 2.4: robertson1983pathwidth
• proof
• Lemma 3.1
• proof
• proof
• Theorem 3.2
• proof