2023/01/04 by Goldberg, Leslie Ann, Roth, Marc
#Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2301.01696
Given a class of graphs H, the problem \oplusSub(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 \oplusSub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)⋅ |G|O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that \oplusSub(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).