Minimum Leaf Out-branching and Related Problems

TL;DR

This paper studies the computational complexity of the Minimum Leaf Out-branching problem in digraphs, providing polynomial solutions for acyclic graphs, NP-hardness results, and a fixed-parameter tractable algorithm with a problem kernel.

Contribution

It proves MinLOB is polynomial for acyclic digraphs, establishes NP-hardness for general cases, and develops an FPT algorithm with a kernel of size O(k^2).

Findings

Polynomial-time solution for acyclic digraphs.

NP-hardness of MinLOB in general digraphs.

An FPT algorithm with a kernel of size O(k^2).

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. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph $D$ of order $n$ and a positive integral parameter $k$, check whether $D$ contains an out-branching with at most $n-k$ leaves (and find such an out-branching if it exists). We find a problem kernel of order $O(k^2)$ and construct an algorithm of running time $O(2^{O(k\log k)}+n^6),$ which is an `additive' FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.