A 4 3 -approximation for the maximum leaf spanning arborescence problem in DAGs

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 $r\in V\left(G\right)$ 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.