We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth. In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through ``requirement shifting''. Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and ``asymptotic'' SETH-based reductions for structural parameters.

翻译：我们研究满足给定顶点度数约束的最小代价生成树计算问题：给定图 $G$ 和约束函数 $D$，我们需要寻找一棵（最小代价）生成树 $T$，使得每个顶点 $v$ 在 $T$ 中的度数满足 $D(v)$ 的指定要求。具体地，我们考虑三种按一般性程度排序的约束函数 —— $D$ 可以为每个顶点分配一个允许的度数列表、度数的上界，或指定的确切度数。通过结合新颖技术与现有最先进的工具，我们获得了这些问题在考虑输入图 $G$ 的最经典图参数时的几乎完整的细粒度复杂性概况。特别地，我们展示了当以路径宽度和割宽度为参数时这些问题的 SETH-紧致上下界，一个以团宽度为参数的 ETH-紧致算法，以及一个接近 SETH-紧致的以树宽度为参数的算法。为了获得关于团宽度的上界，我们开发了一种通过“需求转移”的双重表示新技巧。利用这一技巧，我们还为以团宽度为参数的精确叶生成树问题获得了 ETH-紧致的单指数 XP 算法，这解决了 Cygan 等人 [FOCS 2011, TALG 2022] 经典 Cut and Count 方法中关于团宽度的最后遗留开放情况。这展示了我们技巧的通用性及其对其他问题的潜在适用性。此外，为了建立我们的下界和上界，我们引入了多个可能具有独立意义的工具，包括惰性着色和基于结构参数的“渐近”SETH 归约。