M-convexity of the minimum-cost packings of arborescences
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In this paper, we first prove that the minimum-cost packings of disjoint k arborescences (minimum-cost k-arborescences) induce an M-convex function defined on the integer vectors on the vertex set. This M-convexity provides a short proof for a theorem of Bernáth and Király (SODA 2016) stating that the root vectors of the minimum-cost k-arborescences form a base polyhedron of a submodular function. We then extend this M-convexity to the M^-convexity of the minimum-cost k-branchings. The proof is based on a new theorem on packing disjoint k-branchings, which extends Edmonds' disjoint branchings theorem and is of independent interest. Finally, building upon this M^-convexity, we present a new problem called the prize-collecting k-branching problem, and show that it can be solved in strongly polynomial time if the penalty function is M^-convex.
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