  We investigate a relaxation of the notion of treewidth-fragility, namely
tree-independence-number-fragility. In particular, we obtain polynomial-time
approximation schemes for independent packing problems 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.
