Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond

TL;DR

This paper investigates the computational complexity of evaluating certain counting properties expressed in first-order logic on various sparse graph classes, establishing both algorithmic possibilities and limitations.

Contribution

It extends known evaluation results from bounded expansion to nowhere dense and almost nowhere dense classes, and shows hardness results for further generalizations.

Findings

Efficient evaluation of counting formulas on nowhere dense classes.

Hardness results for almost nowhere dense classes unless FPT=W[1].

Approximation algorithms for counting formulas in almost nowhere dense classes.

Abstract

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.