Abstract
Let us be given a rooted digraph D=(V+s,A) with a designated root vertex s. Edmonds' seminal result [Edmonds, J., Edge-disjoint branchings, in: B. Rustin (ed.) Combinatorial Algorithms, Academic Press, New York, 91–96 (1973] states that D has a packing of k spanning s-arborescences if and only if D has a packing of k (s, t)-paths for all t∈V, where a packing means arc-disjoint subgraphs.Let M be a matroid on the set of arcs leaving s. A packing of (s, t)-paths is called M-based if their arcs leaving s form a base of M while a packing of s-arborescences is called M-based if, for all t∈V, the packing of (s, t)-paths provided by the arborescences is M-based. Durand de Gevigney, Nguyen and Szigeti proved in [Durand de Gevigney, O., V.H. Nguyen, and Z. Szigeti, Matroid-based packing of arborescences, SIAM J. Discrete Math., 27(2013), 567–574] that D has an M-based packing of s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t∈V. Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds' theorem such that each s-arborescence is required to be spanning. Specifically, they conjectured that D has an M-based packing of spanning s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t∈V.We disprove this conjecture in its general form and we prove that the corresponding decision problem is NP-complete. However, we prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids.
Highlights
Packing arborescences, or more generally, packing problems concerning connectivity in directed graphs are fundamental subjects in combinatorial optimization
Theorem 1.1 ([3]) There exists a packing of k spanning s-arborescences in a rooted digraph D = (V + s, A) if and only if there exists a packing of k (s, t)-paths in D for every t ∈ V
There exists an M-based packing of spanning s-arborescences in D if and only if there exists an M-based packing of (s, t)-paths in D for every t ∈ V
Summary
More generally, packing problems concerning connectivity in directed graphs are fundamental subjects in combinatorial optimization. Theorem 1.1 ([3]) There exists a packing of k spanning s-arborescences in a rooted digraph D = (V + s, A) if and only if there exists a packing of k (s, t)-paths in D for every t ∈ V . We are interested in a packing of (s, t)-paths whose root arcs form a base of M.
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