In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied degree-bounded spanning tree problem, as well as a previously studied setting where a chain of cuts is given. We give the first constant-factor approximation algorithm; in particular we show how to obtain a multiplicative violation of the crossing bounds of less than 22 while losing less than a factor of 5 in terms of cost. Our result compares to the natural LP relaxation. As a consequence, our results show that given a $k$-edge-connected graph and a laminar family $\mathcal{L} \subseteq 2^V$ of cuts, there exists a spanning tree which contains only an $O(1/k)$ fraction of the edges across every cut in $\mathcal{L}$. This can be viewed as progress towards the Thin Tree Conjecture, which (in a strong form) states that this guarantee can be obtained for all cuts simultaneously.

翻译：在层状约束生成树问题中，目标是寻找一棵最小成本生成树，使得给定层状族中每个割被跨越的次数满足上界约束。该问题推广了被广泛研究的度约束生成树问题，以及此前研究的链式割约束场景。本文首次给出常数因子近似算法：我们特别展示了如何在成本损失小于5倍的情况下，实现跨越约束的乘法违反因子小于22。该结果与自然线性规划松弛相比较。作为推论，我们的结果表明：给定一个$k$边连通图和一个层状割族$\mathcal{L} \subseteq 2^V$，存在一棵生成树，其对$\mathcal{L}$中每个割的跨越边数仅占该割总边数的$O(1/k)$比例。这可视作向薄树猜想（Thin Tree Conjecture）迈进的进展，该猜想的强形式声称可对所有割同时实现该保证。