We give a number of approximation metatheorems for monotone maximization problems expressible in the first-order logic, in substantially more general settings than the previously known. We obtain * constant-factor approximation algorithm in any class of graphs with bounded expansion, * a QPTAS in any class with strongly sublinear separators, and * a PTAS in any fractionally treewidth-fragile class (which includes all common classes with strongly sublinear separators. Moreover, our tools also give an exact subexponential-time algorithm in any class with strongly sublinear separators.

翻译：我们给出了几种近似元数,用于处理一阶逻辑中比先前已知的更一般得多的逻辑中体现的单调最大化问题。我们获得了在任何类型的图形中具有带宽扩展的常数因素近似算法,在任何类别中具有强分线分离器的QPTAS 和在任何分数的树枝分线分解器类别(包括所有具有强分线分离器的普通类别。此外,我们的工具还给出了在任何类别中具有强分线分离器的精确的亚伸缩时间算法。