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

Neuwohner, Meike ORCID: 0000-0002-3664-3687 (2025) A 4/3-approximation for the maximum leaf spanning arborescence problem in DAGs. Mathematical Programming. ISSN 0025-5610

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Abstract

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

Item Type:	Article
Additional Information:	© 2025 The Author
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	09 May 2025 20:24
Last Modified:	02 Jun 2025 07:12
URI:	http://eprints.lse.ac.uk/id/eprint/128103

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