Algorithm for finding k-vertex out-trees and its application to k-internal out-branching problem

摘要

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 running time O~*(5.704~k) and O~*(6.14~k), 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)[9].