Approximation metatheorems for classes with bounded expansion

Baker's technique is a powerful tool for designing polynomial-time approximation schemes, in particular for all optimization problems expressible in monotone first-order logic. However, it can only be used in rather restricted graph classes. We show that maximization problems expressible in monotone first-order logic admit PTAS under a much weaker assumption of fractional treewidth-fragility, and QPTAS on all hereditary classes with sublinear separators. Moreover, the same technique gives constant-factor approximation for these problems in any class of graphs of bounded expansion.