On Complexity of Minimum Leaf Out-branching Problem
Abstract
Given a digraph , the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer , the problem of checking whether there is an out-branching with at most leaves is polynomial time solvable.
Cite
@article{arxiv.0808.0980,
title = {On Complexity of Minimum Leaf Out-branching Problem},
author = {Peter Dankelmann and Gregory Gutin and Eun Jung Kim},
journal= {arXiv preprint arXiv:0808.0980},
year = {2008}
}