Better algorithms and bounds for directed maximum leaf problems

Abstract

The DIRECTED MAXIMUM LEAF OUT-BRANCHING problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that - every strongly connected digraph D of order n with minimum indegree at least 3 has an out-branching with at least (n/4)1/3 - 1 leaves; - if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph is O(k log k); - it can be decided in time 2O(k log2k)·nO(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. All improvements use properties of extremal structures obtained after applying local search and properties of some out-branching decompositions. © Springer-Verlag Berlin Heidelberg 2007.