Consider the Telephone Broadcast problem in which an input is a connected graph $G$ on $n$ vertices, a source vertex $s \in V(G)$, and a positive integer $t$. The objective is to decide whether there is a broadcast protocol from $s$ that ensures that all the vertices of $G$ get the message in at most $t$ rounds. We consider the broadcast protocol where, in a round, any node aware of the message can forward it to at most one of its neighbors. As the number of nodes aware of the message can at most double at each round, for a non-trivial instance we have $n \le 2^t$. Hence, the brute force algorithm that checks all the permutations of the vertices runs in time $2^{2^{\calO(t)}} \cdot n^{\calO(1)}$. As our first result, we prove this simple algorithm is the best possible in the following sense. Telephone Broadcast does not admit an algorithm running in time $2^{2^{o(t)}} \cdot n^{\calO(1)}$, unless the \ETH\ fails. To the best of our knowledge, this is only the fourth example of \NP-Complete problem that admits a double exponential lower bound when parameterized by the solution size. It also resolves the question by Fomin, Fraigniaud, and Golovach [WG 2023]. In the same article, the authors asked whether the problem is \FPT\ when parameterized by the feedback vertex set number of the graph. We answer this question in the negative. Telephone Broadcast, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is \NP-\complete. We find this a relatively rare example of problems that admit a polynomial-time algorithm on trees but is \NP-\complete\ on graphs of treewidth two.

翻译：考虑电话广播问题，其输入是一个包含 $n$ 个顶点的连通图 $G$、一个源顶点 $s \in V(G)$ 以及一个正整数 $t$。目标是判断是否存在一个从 $s$ 开始的广播协议，使得在最多 $t$ 轮内，$G$ 中的所有顶点都能收到消息。我们考虑的广播协议中，每轮任何已知消息的节点最多可向一个邻居转发消息。由于每轮已知消息的节点数最多翻倍，对于非平凡实例，我们有 $n \le 2^t$。因此，检查所有顶点排列的暴力算法运行时间为 $2^{2^{\calO(t)}} \cdot n^{\calO(1)}$。作为我们的第一个结果，我们证明该简单算法在以下意义上是最优的：电话广播不存在运行时间为 $2^{2^{o(t)}} \cdot n^{\calO(1)}$ 的算法，除非 $\ETH$ 失败。据我们所知，这是第四个以解大小为参数时具有双指数下界的 \NP-完全问题实例。该结果也解决了 Fomin、Fraigniaud 和 Golovach [WG 2023] 提出的问题。在同一篇文章中，作者询问该问题在以图的反馈顶点集数为参数时是否为 \FPT。我们对此问题给出否定回答。当限制在反馈顶点数为 1 的图（即树宽为 2）时，电话广播是 \NP-完全的。我们发现这是一个相对罕见的例子：该问题在树上具有多项式时间算法，但在树宽为 2 的图上却是 \NP-完全的。