AI 中文总结
该论文证明了广播控制数$\gamma_b(G)$至多为多重打包数$mp(G)$的两倍，解决了相关猜想，还得到了最大多重打包问题的多项式时间2-近似算法。
AI 中文摘要
对于顶点集为$V$、边集为$E$的图$G=(V,E)$，函数$f: V \rightarrow \{0,1,2,..., diam(G)\}$被称为$G$上的一个广播。若对每个顶点$u \in V$，都存在$G$中的顶点$v$（允许$u = v$）使得$f(v) > 0$且$d(u, v) \leq f(v)$，则称$f$为$G$上的控制广播。控制广播$f$的代价为所有顶点的$f(v)$之和，控制广播的最小代价即为$G$的广播控制数，记为$\gamma_b(G)$。多重打包是图$G=(V,E)$的顶点子集$M \subseteq V$，满足对每个顶点$v \in V$和每个整数$r \geq 1$，以$v$为中心、半径为$r$的球中最多包含$r$个$M$中的顶点，即距离$v$不超过$r$的$M$中顶点数至多为$r$。$G$的多重打包数是其最大多重打包的基数，记为$mp(G)$。已知$mp(G) \leq \gamma_b(G)$，2014年Hartnell和Mynhardt证明了当$mp(G) \geq 2$时，$\gamma_b(G) \leq 3mp(G)-2$；2019年Beaudou、Brewster和Foucaud将该界改进为$\gamma_b(G) \leq 2 mp(G)+3$，并猜想$\gamma_b(G) \leq 2 mp(G)$。本文证明对所有图$G$，$\gamma_b(G) \leq 2 mp(G)$，解决了该猜想；证明具有构造性，可得到针对最大多重打包问题的多项式时间2-近似算法，改进了此前的近似因子$2+o(1)$。
英文摘要
For a graph $ G = (V, E) $ with a vertex set $ V $ and an 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 dominating broadcast on $ G $. The cost of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $ γ_{b}(G) $. A multipacking is a set $ M \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 $ M $, that is, there are at most $ r $ vertices in $ M $ at a distance at most $ r $ from $ v $ in $ G $. The multipacking number of $ G $ is the maximum cardinality of a multipacking of $ G $ and is denoted by $ mp(G) $. It is known that $mp(G)\leqγ_b(G)$. In 2014, Hartnell and Mynhardt proved that $γ_b(G)\leq 3mp(G)-2$ whenever $mp(G)\geq2$. In 2019, Beaudou, Brewster, and Foucaud improved this bound to $γ_b(G)\leq 2 mp(G)+3$ and conjectured that $γ_b(G)\leq 2 mp(G)$. We solve their conjecture by proving that $γ_b(G)\leq 2 mp(G)$ for every graph $G$. Our proof is constructive and yields a polynomial-time $2$-approximation algorithm for Maximum Multipacking problem which improves the earlier approximation factor $2+o(1)$.
