Better Algorithms and Bounds for Directed Maximum Leaf Problems

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 in-degree 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 outbranching with k leaves, then the pathwidth of its underlying graph is O(k log k); -- it can be decided in time 2O(k log2 k) ċ 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 outbranching decompositions.

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