For a graph $ G = (V, E) $ with vertex set $ V $ and edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $ G $ (possibly, $ u = v $) such that $ f (v) > 0 $ and $ d(u, v) \leq f (v) $, then $ f $ is called a \textit{dominating broadcast} on $ G $. The \textit{cost} of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a dominating broadcast is the \textit{broadcast domination number} of $G$, denoted by $ \gamma_{b}(G) $. A \textit{multipacking} is a set $ S \subseteq V $ in a graph $ G = (V, E) $ such that for every vertex $ v \in V $ and for every integer $ r \geq 1 $, the ball of radius $ r $ around $ v $ contains at most $ r $ vertices of $ S $, that is, there are at most $ r $ vertices in $ S $ at a distance at most $ r $ from $ v $ in $ G $. The \textit{multipacking number} of $ G $ is the maximum cardinality of a multipacking of $ G $ and is denoted by $ mp(G) $. We show that, for any cactus graph $G$, $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for cactus graphs by constructing an infinite family of cactus graphs such that the ratio $\gamma_b(G)/mp(G)=4/3$, with $mp(G)$ arbitrarily large. This result shows that, for cactus graphs, we cannot improve the bound $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ to a bound in the form $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$, for any constant $c_1<4/3$ and $c_2$. Moreover, we provide an $O(n)$-time algorithm to construct a multipacking of $G$ of size at least $ \frac{2}{3}mp(G)-\frac{11}{3} $, where $n$ is the number of vertices of the graph $G$.

翻译：对于图 $G = (V, E)$，其顶点集为 $V$，边集为 $E$，函数 $f : V \rightarrow \{0, 1, 2, . . . , diam(G)\}$ 被称为 $G$ 上的一个*广播*。若对于每个顶点 $u \in V$，存在 $G$ 中的一个顶点 $v$（可能 $u = v$），使得 $f(v) > 0$ 且 $d(u, v) \leq f(v)$，则 $f$ 被称为 $G$ 上的一个*支配广播*。支配广播 $f$ 的*代价*为 $\sum_{v\in V}f(v)$。最小代价的支配广播称为 $G$ 的*广播支配数*，记为 $\gamma_{b}(G)$。*多重打包*是图 $G = (V, E)$ 中的一个顶点集 $S \subseteq V$，满足对于每个顶点 $v \in V$ 和每个整数 $r \geq 1$，以 $v$ 为中心半径为 $r$ 的球包含至多 $r$ 个 $S$ 中的顶点，即在 $G$ 中与 $v$ 距离不超过 $r$ 的 $S$ 中顶点数不超过 $r$ 个。$G$ 的*多重打包数*是 $G$ 中多重打包的最大基数，记为 $mp(G)$。我们证明，对于任意仙人掌图 $G$，有 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$。我们还通过构造一个无穷族仙人掌图，使得比值 $\gamma_b(G)/mp(G)=4/3$ 且 $mp(G)$ 任意大，从而说明对于仙人掌图，$\gamma_b(G)-mp(G)$ 可能任意大。该结果表明，对于仙人掌图，我们无法将界 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ 改进为形如 $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$（其中 $c_1<4/3$，$c_2$ 为任意常数）的形式。此外，我们提供了一种 $O(n)$ 时间的算法，用于构造 $G$ 中大小为至少 $\frac{2}{3}mp(G)-\frac{11}{3}$ 的多重打包，其中 $n$ 是图 $G$ 的顶点数。