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Preprint 46-2007

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

An FPT Algorithm for Directed Spanning k-Leaf
Paul Bonsma and Frederic Dorn
not available
FPT algorithm, maximum leaf, directed graph, spanning tree, out-branching
An out-branching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given digraph D has an out-branching with at least k leaves (Directed Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when k is chosen as the parameter. Previously this was only known for restricted classes of directed graphs.
The main new ingredient in our approach is a lemma that shows that given a locally optimal out-branching of a directed graph in which every arc is part of at least one out-branching, either an out-branching with at least k leaves exists, or a path decomposition with width O(k3) can be found. This enables a dynamic programming based algorithm of running time 2O(k^3 log k) nO(1), where n=|V(D)|.
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