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We give a number of approximation metatheorems for monotone maximizationproblems expressible in the first-order logic, in substantially more generalsettings than the previously known. We obtain * constant-factor approximationalgorithm in any class of graphs with bounded expansion, * a QPTAS in any classwith strongly sublinear separators, and * a PTAS in any fractionallytreewidth-fragile class (which includes all common classes with stronglysublinear separators. Moreover, our tools also give an exactsubexponential-time algorithm in any class with strongly sublinear separators.

Approximation metatheorems for classes with bounded expansion