Parameterized Counting and Cayley Graph Expanders.

Given a graph property phi, we consider the problem \# EDGESUB(phi), where the input is a pair of a graph G and a positive integer k, and the task is to compute the number of k-edge subgraphs in G that satisfy phi. Specifically, we study the parameterized complexity of \#EDGESUB(phi) with respect to both approximate and exact counting, as well as its decision version EDGESUB(phi). Among others, our main result fully resolves the case of minor-closed properties phi: the decision problem EDGESUB(phi) always admits a fixed-parameter tractable algorithm, and the counting problem \#EDGESUB(phi) always admits a fixed-parameter tractable randomized approximation scheme. For exact counting, we present an exhaustive and explicit criterion on the property phi which, if satisfied, yields fixed-parameter tractability and otherwise \#W[1]-hardness. Additionally, our hardness results come with an almost tight conditional lower bound under the exponential time hypothesis. Our main technical result concerns the exact counting problem: Building upon the breakthrough result of Curticapean, Dell, and Marx (Symposium on Theory of Computing 2017), we express the number of subgraphs satisfying phi as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyze the coefficients in the aforementioned linear combinations over the field \BbbFp which gives us significantly more control over the cancelation behavior of the coefficients.