Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs - Bang-Jensen, Jørgen; Saurabh, Saket; Simonsen, Sven (Algorithmica, 2016)
A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer ℓ , we can test whether D contains ℓ arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc - Disjoint Branchings and Non - Disconnecting Out - Branching . In Arc - Disjoint Branchings ( Non - Disconnecting Out - Branching ), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non - Disconnecting Out - Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc - Disjoint Branchings is FPT on strong digraphs. The algorithm for Non - Disconnecting Out - Branching runs in time 2 O ( k ) n O ( 1 ) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.