Minimum leaf out-branching problems

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 k ) and construct an algorithm of running time O(2 O(klogk)∈+∈n 3), which is an 'additive' FPT algorithm. © 2008 Springer-Verlag Berlin Heidelberg.