Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with well-defined properties have been crucial in recent progress. We consider the problem of finding a spanning tree subject to constraints on the edges in cuts forming a laminar family of small width. Our main contribution is a new dynamic programming approach where the value of a table entry does not only depend on the values of previous table entries, as it is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types. In combination with other techniques -- including negatively correlated rounding and a polyhedral approach that, in the problems we consider, allows for avoiding potential losses in the objective through the randomized rounding -- we obtain several new results. We first present a quasi-polynomial time algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an essentially optimal guarantee. More precisely, each chain constraint is violated by a factor of at most $1+\varepsilon$, and the cost is no larger than that of an optimal solution not violating any chain constraint. The best previous procedure is a bicriteria approximation violating each chain constraint by up to a constant factor and losing another factor in the objective. Moreover, our approach can naturally handle lower bounds on the chain constraints, and it can be extended to constraints on cuts forming a laminar family of constant width. Furthermore, we show how our approach can also handle parity constraints (or, more precisely, a proxy thereof) as used in the context of (Path) TSP and one of its generalizations, and discuss implications in this context.

翻译：在各类近似算法中，满足额外约束条件的短生成树是重要的构建模块。特别是在旅行商问题（TSP）背景下，寻找具有良好定义属性的生成树的新技术对近年来的进展至关重要。我们研究在切割约束下寻找生成树的问题，这些切割形成宽度较小的层状族。我们的主要贡献是一种新的动态规划方法，其中表条目的值不仅像通常情况那样依赖于先前表条目的值，还依赖于与每个表条目一起存储的特定代表性解。这使得我们能够处理广泛的约束类型。结合其他技术——包括负相关舍入和一种多面体方法（在我们考虑的问题中，该方法允许通过随机舍入避免目标函数的潜在损失）——我们获得了若干新成果。我们首先提出了最小链约束生成树问题的一个准多项式时间算法，具有本质上最优的保证。更精确地说，每个链约束被违反的因子至多为 $1+\varepsilon$，且成本不超过任何不违反链约束的最优解的成本。先前的最佳方法是双准则近似，每个链约束被违反的因子为常数，且目标函数中还有额外损失。此外，我们的方法能自然地处理链约束的下界，并可扩展到切割形成常宽度层状族的约束。进一步，我们展示了该方法如何应用于（路径）TSP及其一个推广问题中的奇偶约束（或更精确地，其代理约束），并讨论了在这一背景下的启示。