The Maximum Leaf Spanning Arborescence problem (MLSA) in directed acyclic graphs (dags) is defined as follows: Given a directed acyclic 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 in dags is known to be APX-hard as reported by Nadine Schwartges, Spoerhase, and Wolff (Approximation and Online Algorithms, Springer, Berlin Heidelberg, 2012) and the best known approximation guarantee of \(\frac{7}{5}\) is due to Fernandes and Lintzmayer (J. Comput. Syst. Sci. 135: 158–174,2023): They prove that any \(\alpha \) -approximation for the 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 dags, and provide a \(\frac{7}{5}\) -approximation for the hereditary 3-set packing problem. In this paper, we improve upon this result by providing a \(\frac{4}{3}\) -approximation for the hereditary 3-set packing problem, and, thus, the MLSA in dags. The algorithm that we study is a simple local search procedure considering swaps of size up to 10 and can be analyzed via a two-stage charging argument. We further provide a clear picture of the general connection between the MLSA in dags and set packing by rephrasing the MLSA in dags as a hereditary set packing problem. With a much simpler proof, we extend the reduction by Fernandes and Lintzmayer and show that an \(\alpha \) -approximation for the hereditary k-set packing problem implies a \(\max \{\frac{k+1}{k},\alpha \}\) -approximation for the MLSA dags. On the other hand, we provide lower bound examples proving that our approximation guarantee of \(\frac{4}{3}\) is best possible for local search algorithms with constant improvement size.
