Leafy Spanning Arborescences in DAGs

TL;DR

This paper improves approximation algorithms for finding maximum-leaf spanning arborescences in rooted directed acyclic graphs, a problem relevant for efficient broadcasting in networks, and establishes inapproximability bounds for weighted variants.

Contribution

It presents a (3/2)-approximation algorithm for the problem on rooted DAGs, improving the previous 2-approximation, and extends inapproximability results to the weighted case.

Findings

Improved approximation ratio from 2 to 1.5 for the problem.

Derived inapproximability bounds for the vertex-weighted version.

Adapted undirected case results to directed acyclic graphs.

Abstract

Broadcasting in a computer network is a method of transferring a message to all recipients simultaneously. It is common in this situation to use a tree with many leaves to perform the broadcast, as internal nodes have to forward the messages received, while leaves are only receptors. We consider the subjacent problem of, given a directed graph~$D$, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. In this paper, we concentrate on the class of rooted directed acyclic graphs, for which the problem is known to be MaxSNP-hard. A 2-approximation was previously known for this problem on this class of directed graphs. We improve on this result, presenting a (3/2)-approximation. We also adapt a result for the undirected case and derive an inapproximability result for the vertex-weighted version of Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs.