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 outbranching with at most n − k leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k · 2k ) and construct an algorithm of running time O(2O(k log k) + n3 ), which is an ‘additive’ FPT algorithm.

• Type: Conference
• Year: 2008
• Where: AAIM
• Authors: Gregory Gutin, Igor Razgon, Eun Jung Kim
• Keywords: AAIM 2008, Acyclic Digraphs, Algorithms, Leaf Out-Branching Problem, Minimum Possible Number