On Complexity of Minimum Leaf Out-branching Problem

TL;DR

This paper explores the computational complexity of the Minimum Leaf Out-Branching problem in directed graphs, showing it is NP-hard for certain width parameters but fixed-parameter tractable when these parameters are restricted.

Contribution

It extends known polynomial cases by proving NP-hardness for digraphs with width 1 and establishing fixed-parameter tractability for bounded width digraphs.

Findings

NP-hardness for digraphs with width 1

Polynomial-time algorithms for fixed k and restricted width

Extension of polynomiality results to broader classes

Abstract

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ 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 $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable.