  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 $c:A\to \mathbb{R}$, 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 $w:A\to \mathbb{R}_+$ (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.
