A $\frac{4}{3}$-Approximation for the Maximum Leaf Spanning Arborescence Problem in DAGs

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.