Double Exponential Lower Bound for Telephone Broadcast

TL;DR

This paper establishes a double exponential lower bound for the Telephone Broadcast problem, showing that the brute force approach is essentially optimal under ETH, and proves the problem's NP-completeness even on graphs with feedback vertex set one.

Contribution

It proves a tight double exponential lower bound for the problem's complexity and answers open questions about its fixed-parameter tractability and complexity on restricted graph classes.

Findings

Brute force algorithm is optimal under ETH.

Telephone Broadcast is NP-complete on graphs with feedback vertex set one.

The problem is polynomial-time solvable on trees but NP-complete on graphs with feedback vertex set one.

Abstract

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.