Double Exponential Lower Bound for Telephone Broadcast

TL;DR

This article proves Telephone Broadcast, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is \NP-\complete, when restricted to graphs of the feedback vertex set number of the graph.

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 Ð 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.

Figures (4)

• Figure 1: An application of Observation \ref{['obs:message-propelling']}. For an edge $(u, v)$, the integer $t'$ associated with it denotes that at time $t'$, vertex $u$ forwarded message to vertex $v$.
• Figure 2: Overview of the reduction in Section \ref{['sec:eth-lower-bound']}. For clarity, the figure only shows vertices corresponding to variables $v^1_1$ and $v^2_1$ (part of $v^2_2$ but not $v^3_1$). With renaming the variables, $C_1 \equiv (\neg v^1_1 \lor \neg v^2_1 \lor v^3_1)$, $C_2 \equiv (v^1_1 \lor v^2_1 \lor v^2_2)$, and $C_3 \equiv (v^1_1 \lor \neg v^2_1)$. $C_2$ and $C_3$ are connected to $y^1_1$ and $z^1_1$, respectively, to encode the first positive and the second positive appearance of variable $v^1_1$. The thick line with an arrow towards $\delta$-type vertices denotes the long paths whose length is adjusted so that Observation \ref{['obs:message-propelling']} is applicable. For this example, suppose $t = 13$. The blue and green colored time-stamps show a broadcasting protocol that forwards the messages to $\neg x^1_1$ (before forwarding it to $x^1_1$) and to $x^2_2$ (before forwarding it to $\neg x^2_2$), respectively. This corresponds to assigning $v^1_1 = \texttt{False}$ and $v^{2}_1 = \texttt{True}$. These choices ensure that vertices $c'_2$ and $c'_3$ get the message in time which translate to satisfying clauses $C_2$ and $C_3$. The red and purple colored time stamps, respectively, show that the message forwarded by $x^1_1$ or $\neg x^2_1$ can not reach the clause vertices. In the case of red time stamps, the message can't reach even if it deviates from others.
• Figure 3: Illustration of a pair of vertices in bucket $B^{\ell^{\circ}}$. For the sake of clarity, we do not show all the vertices. For Lemma \ref{['lemma:eth-lower-bound-forward']}, the blue time stamps show the round in which a vertex forwards the message along the edge. For Lemma \ref{['lemma:eth-lower-bound-backward']}, the green-shaded path and red-shaded path show the message propelled along the path due to constraints imposed by $\delta^{\ell^{\circ}}_i$ and $\delta^{\ell^{\circ} +1}_{i'}$, respectively.
• Figure 4: Overview of the reduction. The thick line shows long paths of the specified size.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4: Message Propelling and Downtime
• Proposition 6
• Lemma 7
• Lemma 8
• Lemma 11
• Lemma 12
• Lemma 13