Courcelle's theorem and its adaptations to cliquewidth have shaped the field of exact parameterized algorithms and are widely considered the archetype of algorithmic meta-theorems. In the past decade, there has been growing interest in developing parameterized approximation algorithms for problems which are not captured by Courcelle's theorem and, in particular, are considered not fixed-parameter tractable under the associated widths. We develop a generalization of Courcelle's theorem that yields efficient approximation schemes for any problem that can be captured by an expanded logic we call Blocked CMSO, capable of making logical statements about the sizes of set variables via so-called weight comparisons. The logic controls weight comparisons via the quantifier-alternation depth of the involved variables, allowing full comparisons for zero-alternation variables and limited comparisons for one-alternation variables. We show that the developed framework threads the very needle of tractability: on one hand it can describe a broad range of approximable problems, while on the other hand we show that the restrictions of our logic cannot be relaxed under well-established complexity assumptions. The running time of our approximation scheme is polynomial in $1/\varepsilon$, allowing us to fully interpolate between faster approximate algorithms and slower exact algorithms. This provides a unified framework to explain the tractability landscape of graph problems parameterized by treewidth and cliquewidth, as well as classical non-graph problems such as Subset Sum and Knapsack.

翻译：库尔塞勒定理及其对团宽度的适配塑造了精确参数化算法领域，并被广泛视为算法元定理的原型。过去十年间，针对库尔塞勒定理未涵盖的问题——尤其是那些被认为在相关宽度下不具备固定参数可追踪性的问题——开发参数化近似算法的兴趣日益增长。我们提出库尔塞勒定理的一个推广，该推广能为任何可通过扩展逻辑“受限CMS O”（Blocked CMSO）描述的问题生成高效近似方案。该逻辑通过所谓的权重比较，能够对集合变量的大小进行逻辑陈述。它通过涉及变量的量词交替深度来控制权重比较：允许零交替变量进行全面比较，而对单交替变量进行有限比较。我们证明，所提出的框架恰好把握了可处理性的关键：一方面，它能描述广泛的近似问题；另一方面，我们表明，在公认的复杂性假设下，我们的逻辑限制无法被放宽。我们的近似方案运行时间关于$1/\varepsilon$为多项式，从而能完全在更快的近似算法和更慢的精确算法之间进行插值。这为解释以树宽和团宽度参数化的图问题以及诸如子集和与背包问题等经典非图问题的可处理性格局提供了统一框架。