Parameterised and fine-grained subgraph counting, modulo 2

Journal article

Given a class of graphs H, the problem ⊕Sub(H) is defined as follows. The input is a graph H ∈ 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 H the problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|) · |G| O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ 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 H, and (II) all tree pattern classes, i.e., all classes H such that every H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

Authors: Goldberg, LA; Roth, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Algorithmica
• Volume: 86
• Issue: 4
• Pages: 944–1005
• Publication date: 2023-11-02
• Acceptance date: 2023-10-10
• DOI: 10.1007/s00453-023-01178-0
• EISSN: 1432-0541
• ISSN: 0178-4617
• Language: English
• Keywords: parameterised complexity; subgraph counting; modular counting; fine-grained complexity