Parameterised and Fine-grained Subgraph Counting, modulo $2$

TL;DR

This paper characterizes when counting subgraphs modulo 2 is fixed-parameter tractable for hereditary and tree pattern classes, confirming a conjecture under the randomized ETH and providing tight bounds.

Contribution

It proves the conjecture that al Sub(H) is FPT iff al H is matching splittable for hereditary and tree classes, assuming the randomized ETH.

Findings

Confirmed the conjecture for hereditary pattern classes.

Confirmed the conjecture for tree pattern classes.

Established tight bounds for hereditary patterns.

Abstract

Given a class of graphs $\mathcal{H}$, the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is defined as follows. The input is a graph $H\in \mathcal{H}$ together with an arbitrary graph $G$. The problem is to compute, modulo $2$, the number of subgraphs of $G$ that are isomorphic to $H$. The goal of this research is to determine for which classes $\mathcal{H}$ the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is fixed-parameter tractable (FPT), i.e., solvable in time $f(|H|)\cdot |G|^{O(1)}$. Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $\oplus\mathsf{Sub}(\mathcal{H})$ is FPT if and only if the class of allowed patterns $\mathcal{H}$ is "matching splittable", which means that for some fixed $B$, every $H \in \mathcal{H}$ can be turned into a matching (a graph in which every vertex has degree at most $1$) by removing at most $B$ vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes $\mathcal{H}$, and (II) all tree pattern classes, i.e., all classes $\mathcal{H}$ such that every $H\in \mathcal{H}$ is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).