Parameterised and Fine-Grained Subgraph Counting, Modulo 2

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 pr...

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Bibliographic Details
Published in:Algorithmica Vol. 86; no. 4; pp. 944 - 1005
Main Authors: Goldberg, Leslie Ann, Roth, Marc
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2024
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph matching
Graph theory
Lower bounds
Mathematics of Computing
Theory of Computation
Theory of computation
Problems
Subgraph counting
Fine-grained complexity
Parameterised complexity
reductions and completeness Mathematics of computing
Discrete mathematics
Modular counting
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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).
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-023-01178-0