Blocking Optimal Arborescences

Abstract

The problem of covering minimum cost common bases of two matroids is NP-complete, even if the two matroids coincide, and the costs are all equal to 1. In this paper we show that the following special case is solvable in polynomial time: given a digraph D = (V,A) with a designated root node r ∈ V and arc-costs c:A → ℝ, find a minimum cardinality subset H of the arc set A such that H intersects every minimum c-cost r-arborescence. The algorithm we give solves a weighted version as well, in which a nonnegative weight function w:A → ℝ₊ is also given, and we want to find a subset H of the arc set such that H intersects every minimum c-cost r-arborescence, and w(H) is minimum. The running time of the algorithm is O(n ³ T(n,m)), where n and m denote the number of nodes and arcs of the input digraph, and T(n,m) is the time needed for a minimum s − t cut computation in this digraph. A polyhedral description is not given, and seems rather challenging.