The weighted $3$-Set Packing problem is defined as follows: As input, we are given a collection $\mathcal{S}$ of sets, each of cardinality at most $3$ and equipped with a positive weight. The task is to find a disjoint sub-collection of maximum total weight. Already the special case of unit weights is known to be NP-hard, and the state-of-the-art are $\frac{4}{3}+\epsilon$-approximations by Cygan and F\"urer and Yu. In this paper, we study the $2$-$3$-Set Packing problem, a generalization of the unweighted $3$-Set Packing problem, where our set collection may contain sets of cardinality $3$ and weight $2$, as well as sets of cardinality $2$ and weight $1$. Building upon the state-of-the-art works in the unit weight setting, we manage to provide a $\frac{4}{3}+\epsilon$-approximation also for the more general $2$-$3$-Set Packing problem. We believe that this result can be a good starting point to identify classes of weight functions to which the techniques used for unit weights can be generalized. Using a reduction by Fernandes and Lintzmayer, our result further implies a $\frac{4}{3}+\epsilon$-approximation for the Maximum Leaf Spanning Arborescence problem (MLSA) in rooted directed acyclic graphs, improving on the previously known $\frac{7}{5}$-approximation by Fernandes and Lintzmayer. By exploiting additional structural properties of the instance constructed in their reduction, we can further get the approximation guarantee for the MLSA down to $\frac{4}{3}$. The MLSA has applications in broadcasting where a message needs to be transferred from a source node to all other nodes along the arcs of an arborescence in a given network.

翻译：加权$3$-集合打包问题定义如下：输入为一个集合族$\mathcal{S}$，其中每个集合的基数至多为$3$并配有正权重。任务目标是找到总权重最大的不相交子集族。即使是单位权重的特例也被已知为NP难问题，当前最优结果由Cygan、F\"urer和Yu提供，为$\frac{4}{3}+\epsilon$近似算法。本文研究$2$-$3$-集合打包问题，它是无权$3$-集合打包问题的一种推广，其中集合族可能包含基数为$3$、权重为$2$的集合，以及基数为$2$、权重为$1$的集合。基于单位权重设置下的最新研究成果，我们成功为更一般的$2$-$3$-集合打包问题给出了$\frac{4}{3}+\epsilon$近似算法。我们相信这一结果可以作为识别可推广单位权重技术的权重函数类别的良好起点。利用Fernandes与Lintzmayer的归约，我们的结果进一步为有根有向无环图中的最大叶生成树状图问题（MLSA）提供了$\frac{4}{3}+\epsilon$近似算法，改进了此前Fernandes与Lintzmayer给出的$\frac{7}{5}$近似。通过挖掘他们归约中构造实例的额外结构性质，我们还能将MLSA的近似保证进一步降低至$\frac{4}{3}$。MLSA在广播领域具有应用价值，其中消息需沿给定网络中树状图的弧从源节点传输至所有其他节点。