  We consider the bilevel minimum spanning tree (BMST) problem where the leader
and the follower choose a spanning tree together, according to different
objective functions. By showing that this problem is NP-hard in general, we
answer an open question stated in by Shi et al. We prove that BMST remains hard
even in the special case where the follower only controls a matching. Moreover,
by a polynomial reduction from the vertex-disjoint Steiner trees problem, we
give some evidence that BMST might even remain hard in case the follower
controls only few edges. On the positive side, we present a polynomial-time
$(|V|-1)$-approximation algorithm for BMST, where $|V|$ is the number of
vertices in the input graph. Moreover, considering the number of edges
controlled by the follower as parameter, we show that 2-approximating BMST is
fixed-parameter tractable and that, in case of uniform costs on leader's edges,
even solving BMST exactly is fixed-parameter tractable. We finally consider
bottleneck variants of BMST and settle the complexity landscape of all
combinations of sum or bottleneck objective functions for the leader and
follower, for the optimistic as well as the pessimistic setting.
