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 $r\in V$ and arc-costs $c:A\rightarrow \mathbb {R}$, find a minimum cardinality subset H of the arc set A such that H intersects every minimum c-cost r-arborescence. By an r-arborescence we mean a spanning arborescence of root r. The algorithm we give solves a weighted version as well, in which a nonnegative weight function $w:A\rightarrow \mathbb {R}_+$ (unrelated to c) 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 $w(H)=\sum _{a\in H}w(a)$ is minimum. The running time of the algorithm is $O(n^3T(n,m))$, 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.