We investigate a relaxation of the notion of fractional treewidth-fragility, namely fractional tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for meta-problems such as finding a maximum-weight sparse induced subgraph satisfying a given $\mathsf{CMSO}_2$ formula on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth, and its applications to exact subexponential-time algorithms.

翻译：本文研究了分数树宽脆弱性概念的一个松弛版本，即分数树独立数脆弱性。特别地，我们针对分数树独立数脆弱图类上的元问题（例如在给定$\mathsf{CMSO}_2$公式约束下寻找最大权稀疏诱导子图）获得了多项式时间近似方案。本文方法统一并扩展了多个看似无关图类上的已知多项式时间近似方案，例如固定维度中胖物体相交图类或真极小封闭图类。我们还研究了分层树独立数的相关概念（分层树宽的一个松弛版本）及其在精确次指数时间算法中的应用。