The Maximum Leaf Spanning Arborescence problem (MLSA) is defined as follows: Given a directed graph $G$ and a vertex $r\in V(G)$ from which every other vertex is reachable, find a spanning arborescence rooted at $r$ maximizing the number of leaves (vertices with out-degree zero). The MLSA has applications in broadcasting, where a message needs to be transferred from a source vertex to all other vertices along the arcs of an arborescence in a given network. In doing so, it is desirable to have as many vertices as possible that only need to receive, but not pass on messages since they are inherently cheaper to build. We study polynomial-time approximation algorithms for the MLSA. For general digraphs, the state-of-the-art is a $\min\{\sqrt{\mathrm{OPT}},92\}$-approximation. In the (still APX-hard) special case where the input graph is acyclic, the best known approximation guarantee of $\frac{7}{5}$ is due to Fernandes and Lintzmayer: They prove that any $\alpha$-approximation for the \emph{hereditary $3$-set packing problem}, a special case of weighted $3$-set packing, yields a $\max\{\frac{4}{3},\alpha\}$-approximation for the MLSA in acyclic digraphs (dags), and provide a $\frac{7}{5}$-approximation for the hereditary $3$-set packing problem. In this paper, we obtain a $\frac{4}{3}$-approximation for the hereditary $3$-set packing problem, and, thus, also for the MLSA in dags. In doing so, we manage to leverage the full potential of the reduction provided by Fernandes and Lintzmayer. The algorithm that we study is a simple local search procedure considering swaps of size up to $10$. Its analysis relies on a two-stage charging argument.

翻译：最大叶生成树形图问题定义如下：给定一个有向图$G$和一个顶点$r\in V(G)$，使得从$r$出发可以到达所有其他顶点，目标是找到一棵以$r$为根的生成树形图，使得其中叶子节点（出度为零的顶点）的数量最大化。该问题在广播通信中具有应用价值，即需要将消息从源顶点通过给定网络的树形图弧线传输至所有其他顶点。在此过程中，我们希望尽可能多地让顶点仅接收消息而无需转发，因为这类节点的构建成本通常更低。本文研究最大叶生成树形图问题的多项式时间近似算法。对于一般有向图，目前最先进的算法具有$\min\{\sqrt{\mathrm{OPT}},92\}$的近似比。在输入图为无环图（该特例仍为APX难问题）的情况下，Fernandes和Lintzmayer提出的算法达到了目前已知最优的近似比$\frac{7}{5}$：他们证明了针对遗传性3-集合打包问题（加权3-集合打包问题的特例）的任意$\alpha$近似算法，均可转化为有向无环图中最大叶生成树图问题的$\max\{\frac{4}{3},\alpha\}$近似算法，并为遗传性3-集合打包问题提供了$\frac{7}{5}$近似算法。本文通过研究规模不超过10的局部交换操作，提出了一种简单的局部搜索算法，并利用两阶段计费论证进行分析，最终为遗传性3-集合打包问题获得了$\frac{4}{3}$近似比，从而也将有向无环图中最大叶生成树形图问题的近似比提升至$\frac{4}{3}$。这一成果充分挖掘了Fernandes和Lintzmayer所提归约方法的潜力。