The $2$-$3$-Set Packing problem and a $\frac{4}{3}$-approximation for the Maximum Leaf Spanning Arborescence problem in rooted dags

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 $\frac{4}{3}+ε$-approximations by Cygan and Fürer 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 $\frac{4}{3}+ε$-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 $\frac{4}{3}+ε$-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.