Polynomial-time approximation schemes for induced subgraph problems on fractionally tree-independence-number-fragile graphs

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.