How heavy independent sets help to find arborescences with many leaves in DAGs

TL;DR

This paper presents a new approximation algorithm for the Maximum Leaf Spanning Arborescence problem in rooted directed acyclic graphs, leveraging a relation to maximum weight set packing to improve approximation ratios.

Contribution

It introduces a novel relation between maximum leaf spanning arborescences and maximum weight set packing, leading to a better 7/5-approximation for the problem.

Findings

Achieved a 7/5-approximation ratio, improving the previous 3/2-approximation.

Established a connection between arborescence problems and set packing.

Potential to enhance approximation algorithms for weighted k-set packing.

Abstract

Trees with many leaves have applications on broadcasting, which is a method in networks for transferring a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology, because they have to forward the messages received. We address a problem that captures the main goal, which is to find spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. This problem is known to be NP-hard in general and MaxSNP-hard on the class of rooted directed acyclic graphs. In this paper, we explore a relation between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation. The approach used might lead to improvements on the best approximation ratios for the weighted k-set packing problem.