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

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.