Broadcast Graph Is NP-complete

TL;DR

This work investigates the computational complexity of the Broadcast Graph problem, which asks whether a graph $G=(V,E)$ has broadcast time from any vertex equal to $\lceil \log n \rceil$ with $n=|V|$. It proves Broadcast Graph is NP-complete by a reduction from ST-Broadcast Time, leveraging a construction that fuses Knödel graphs and binomial trees via a compounding technique to create yes-instances and no-instances. Additionally, it analyzes BC-Size, showing it lies in $\Delta^2_p$ and is $D^P$-hard through a reduction from Unique-SAT, thereby placing BC-Size within a narrow complexity interval between $D^P$-hardness and $\Delta^2_p$-completeness. Overall, the results clarify the difficulty of recognizing optimal broadcast topologies and identifying broadcast centers, with implications for designing efficient one-to-all dissemination networks in graphs.

Abstract

The broadcast model is widely used to describe the process of information dissemination from a single node to all nodes within an interconnected network. In this model, a graph represents the network, where vertices correspond to nodes and edges to communication links. The efficiency of this broadcasting process is evaluated by the broadcast time, the minimum discrete time units required to broadcast from a given vertex. Determining the broadcast time is referred to as the problem Broadcast Time. The set of vertices with the minimum broadcast time among the graph is called the broadcast center. Identifying this center or determining its size are both proven to be NP-hard. For a graph with n vertices, the minimum broadcast time is at least ceil(log2 n). The Broadcast Graph problem asks in a graph of n vertices, whether the broadcast time from any vertex equals ceil(log2 n). Extensive research over the past 50 years has focused on constructing broadcast graphs, which are optimal network topologies for one-to-all communication efficiency. However, the computational complexity of the Broadcast Graph problem has rarely been the subject of study. We believe that the difficulty lies in the mapping reduction for an NP-completeness proof. Consequently, we must construct broadcast graphs for yes-instances and non-broadcast graphs for no-instances. The most closely related result is the NP-completeness of Broadcast Time proved by Slater et al. in 1981. More recently, Fomin et al. has proved that Broadcast Time is fixed-parameter tractable. In this paper, we prove that Broadcast Graph is NP-complete by proving a reduction from Broadcast Time. We also improve the results on the complexity of the broadcast center problem. We show Broadcast Center Size is in delta^2_p, and is DP-hard, implying a complexity upper bound of delta^2_p-complete and a lower bound of DP-hard.

Figures (5)

• Figure 1: binomial trees of degree $0$ to $3$. Each vertex is labeled by its binary representation, This notation allows us to sort the leaves (vertices ends with '0') by their distances to the root in descending order, which is useful in Section 4. In $BT_i$, the first $i-1$ bit of the leaves are considered. For example, in $BT_3$, the first two bits of the leaves are $bin(00)=dec(0)$, $bin(01)=dec(1)$, $bin(10)=dec(2)$, $bin(11)=dec(3)$.
• Figure 2: Example construction of $G_u$ on $G_s$. The dotted triangles represent the binomial trees $T_1, \cdots, T_5$ of degree $\lceil\log{n_s}\rceil+1$, and $T_6$ of degree $\lceil\log{n_s}\rceil$. A Knödel graph $KG_6$ is formed by $r_1, \cdots, r_5$, and $v_s$. The dashed lines represent edges added in step 5 and 6 in Definition 6. The dot-dashed line is $(v_s,r_6)$.
• Figure 3: An example of the construction of $G$. The binomial tree $H^{\hat{x_i}}$ represents the assignment $\hat{x_i}=1$. Edges between $H$'s and $T$'s show the two possible assignments for each variable. Each binomial tree $T^{\hat{x_i}}$ presents a literal $\hat{x_i}$. Each $\delta_i$ is a clause. The binomial tree $T^r$ ensures $s$ and $r$ are in the broadcast center. The path $p_0,\cdots$ forces all vertices other than $s,r,r_{x_{0}},\cdots$ are out of the broadcast center.
• Figure 4: An example for variable, literal, and clause gadgets. Assume $\delta_1=x_0\vee \overline{x_1}\vee\cdots$ and $\delta_2=\overline{x_0}\vee x_1\vee\cdots$ are two clauses in $\phi$. Vertex $\delta_1$ is adjacent to one leaf in $T^{x_0}$ and another leaf in $T^{\overline{x_1}}$ (same for $\delta_2$). If all leaves of $H^{\overline{x_1}}$ are informed at time unit $d_1+1$, then directed edges and time units (in red) show the calls from $l_0^{\overline{x_1}}$ and $l_1^{\overline{x_1}}$ to $\delta_1$ and $\delta_2$ respectively at $d_1+d_2+3$, mimicking the assignment $x_0=False$ and $x_1=False$ to satisfy both $\delta_1$ and $\delta_2$.
• Figure A1: An example of the construction of $G_f$ respected to the $3DM$ instance $(X=\{x_1,x_2,x_3\}$, $Y=\{y_1,y_2,y_3\}$, $Z=\{z_1,z_2,z_3\}$, $W=\{(x_1,y_1,z_1)$,$(x_2,y_2,z_2)$,$(x_3,y_3,z_3)$,$(x_3,y_2,z_1)\},k=3)$. The grey vertex is the originator $v_0$.

Theorems & Definitions (23)

• definition 1
• definition 2
• lemma 1
• proof
• definition 3
• lemma 2
• proof
• theorem 1
• lemma 3
• proof