  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.
