On Finding Directed Trees with Many Leaves

TL;DR

This paper introduces combinatorial bounds and approximation algorithms for the Rooted Maximum Leaf Outbranching problem, improving approximation factors and kernel size in directed spanning trees with many leaves.

Contribution

It presents new combinatorial bounds, a constant factor approximation algorithm, and a quadratic kernel for the problem, advancing the state of the art.

Findings

Constant factor approximation algorithm developed

Quadratic kernel for the problem established

Improved bounds over previous cubic kernel

Abstract

The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ leaves. We use the notion of $s-t$ numbering to exhibit combinatorial bounds on the existence of spanning directed trees with many leaves. These combinatorial bounds allow us to produce a constant factor approximation algorithm for finding directed trees with many leaves, whereas the best known approximation algorithm has a $\sqrt{OPT}$-factor. We also show that Rooted Maximum Leaf Outbranching admits a quadratic kernel, improving over the cubic kernel given by Fernau et al.