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 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 $(n-1)$-approximation algorithm for BMST, where $n$ 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.

翻译：我们考虑的是双层最小横贯树(BMST)问题,即领导人和追随者根据不同的客观功能共同选择横贯树(BMST)的问题。通过显示这个问题一般是NP硬的,我们回答什等人提出的一个未决问题。我们证明,即使是在跟踪者只控制匹配的特殊情况下,BMSST仍然是困难的。此外,通过从脊椎分解的施泰纳树问题中减少二位数,我们提供了一些证据,表明BMST在跟踪者控制少数边缘的情况下甚至可能仍然很困难。在积极的一面,我们为BMST展示了一个多盘时美元(n-1美元-美元-Approcolm)的算法,在这种算法中,美元是输入图中的脊椎数。此外,考虑到由追随者控制的边缘数作为参数,我们表明,2位相位相近的BMST(BST)是固定的参数,而且,如果领导人的边缘控制费用统一,甚至解决BMST(BMS)的精确度是固定的可伸缩的。我们最后认为,制式模式的组合是制式和制式组合。