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 $\textit{broadcast}$ 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 $G = (V, E)$，其中 $V$ 为顶点集，$E$ 为边集，函数 $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ 称为 $G$ 上的一个广播。若对每个顶点 $u \in V$，存在 $G$ 中的某个顶点 $v$（允许 $u = v$）满足 $f(v) > 0$ 且 $d(u, v) \leq f(v)$，则 $f$ 称为 $G$ 上的支配广播。支配广播 $f$ 的成本为 $\sum_{v\in V}f(v)$。支配广播的最小成本即为 $G$ 的广播支配数，记作 $\gamma_{b}(G)$。多重打包是图 $G = (V, E)$ 中的一个顶点子集 $S \subseteq V$，满足：对每个顶点 $v \in V$ 和每个整数 $r \geq 1$，以 $v$ 为中心的半径为 $r$ 的球内至多包含 $S$ 中的 $r$ 个顶点，即在 $G$ 中与 $v$ 距离不超过 $r$ 的 $S$ 中顶点数至多为 $r$。$G$ 的多重打包数定义为 $G$ 中多重打包的最大基数，记作 $mp(G)$。我们证明：对任意仙人掌图 $G$，有 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$。同时，通过构造一个无穷仙人掌图族（其中 $mp(G)$ 任意大且比值 $\gamma_b(G)/mp(G)=4/3$），我们说明对仙人掌图而言，$\gamma_b(G)-mp(G)$ 可以任意大。这一结果表明：对于仙人掌图，无法将界 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ 改进为形如 $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$ 的界，其中 $c_1<4/3$ 且 $c_2$ 为任意常数。此外，我们给出了一个 $O(n)$ 时间算法来构造 $G$ 的一个多重打包，其大小至少为 $\frac{2}{3}mp(G)-\frac{11}{3}$，其中 $n$ 为图 $G$ 的顶点数。