AI 中文总结
本文研究受局部度约束的最小生成树问题，发现指定度最小生成树、有界度最小生成树与度集合最小生成树在有界树深图上的复杂度存在根本差异，前两者固定参数可处理，后者仍W[1]-难。
AI 中文摘要
我们研究受局部度约束的最小生成树计算问题。近期研究（ICALP 2026）表明，该问题的三种自然形式化表述在标准结构图参数（包括树宽、路径宽和团宽）下具有完全相同的参数化复杂度，适用于每个顶点有单个目标度的「指定度最小生成树（Specified Degree MST）」、有度上限的「有界度最小生成树（Bounded Degree MST）」，或配备一组允许度的「度集合最小生成树（Set of Degrees MST）」这三种情形。本文在更严格的参数化设定下研究这些问题，发现它们在有界树深图上的复杂度特征存在根本差异。具体而言，我们证明前两个问题在以输入图的树深为参数时是固定参数可处理的；相比之下，我们表明度集合最小生成树即使结合反馈顶点数（即到树宽1的删除距离），以树深为参数时仍保持W[1]-难。最后，我们表明这种差异似乎是树深特有的：我们排除了度集合最小生成树相对于顶点覆盖数的类似W[1]-难结果，同时也排除了前两个问题相对于到常数路径宽的删除距离的固定参数算法。
英文摘要
We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.
CommentsTo appear at IPEC 2026. This is the original submitted version - reviewer comments will be incorporated in a later version
