Journal Article

A \\nicefrac 43 43-approximation for the maximum leaf spanning arborescence problem in DAGs

Neuwohner, Meike

2025

Overview

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∈ V(G) r∈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 (7/5) 75 is due to Fernandes and Lintzmayer (J. Comput. Syst. Sci. 135: 158–174,2023): They prove that any α α-approximation for the hereditary 3-set packing problem, a special case of weighted 3-set packing, yields a \\max {(4/3),α } max43,α-approximation for the MLSA in dags, and provide a (7/5) 75-approximation for the hereditary 3-set packing problem. In this paper, we improve upon this result by providing a (4/3) 43-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 α α-approximation for the hereditary k-set packing problem implies a \\max {((k+1)/k),α } maxk+1k,α-approximation for the MLSA dags. On the other hand, we provide lower bound examples proving that our approximation guarantee of (4/3) 43 is best possible for local search algorithms with constant improvement size.

• Subject: Full Length Paper