Algorithm for Finding $k$-Vertex Out-trees and its Application to $k$-Internal Out-branching Problem

TL;DR

This paper introduces algorithms to efficiently determine the presence of specific out-trees with k vertices in directed graphs, and applies these methods to solve the k-internal out-branching problem, answering an open question.

Contribution

It presents both randomized and deterministic algorithms with exponential runtime bounds for detecting out-trees, and provides a deterministic solution for the k-internal out-branching problem.

Findings

Deterministic algorithm runs in O^*(5.704^{k(1+o(1))}) time.

Application of the algorithm solves the k-internal out-branching problem.

Answers an open question by Gutin, Razgon, and Kim.

Abstract

An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08).