It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-$r$ minors have constant density. More precisely, the formulas are $\exists x_1 ... x_k \#y \varphi(x_1,...,x_k, y)>N$, where $\varphi$ is an FO-formula. If $\varphi$ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-$r$ clique minors is subpolynomial, is impossible unless FPT=W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({<}) but not FOC1(P). In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.

翻译：已知带有某些计数扩展的一阶逻辑可以在具有有界扩张的图类上高效评估，其中深度为$r$的次图具有恒定密度。更精确地说，公式形式为$\exists x_1 ... x_k \#y \varphi(x_1,...,x_k, y)>N$，其中$\varphi$是一个FO公式。若$\varphi$无量词，则可将此结果推广至无处稠密的图类，并实现近线性FPT运行时间。将此结果进一步推广至更一般的图类（即几乎无处稠密类），其中深度为$r$的团次图大小为次多项式，是不可能的，除非FPT=W[1]。另一方面，在几乎无处稠密类中，我们可以用较小的加法误差逼近此类计数公式。注意这些计数公式包含在FOC({<})中，但不包含在FOC1(P)中。特别地，这意味着部分覆盖问题（如部分支配集）在无处稠密图类上具有近线性运行时间的固定参数算法。