  We give polynomial-time approximation schemes for monotone maximization
problems expressible in terms of distances (up to a fixed upper bound) and
efficiently solvable in graphs of bounded treewidth. These schemes apply in all
fractionally treewidth-fragile graph classes, a property that is true for many
natural graph classes with sublinear separators. We also provide
quasipolynomial-time approximation schemes for these problems in all classes
with sublinear separators.
