Parameterised and Fine-grained Subgraph Counting, modulo 2

01/04/2023
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by   Leslie Ann Goldberg, et al.
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Given a class of graphs โ„‹, the problem โŠ•๐–ฒ๐—Ž๐–ป(โ„‹) is defined as follows. The input is a graph Hโˆˆโ„‹ 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 โ„‹ the problem โŠ•๐–ฒ๐—Ž๐–ป(โ„‹) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)ยท |G|^O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that โŠ•๐–ฒ๐—Ž๐–ป(โ„‹) is FPT if and only if the class of allowed patterns โ„‹ is "matching splittable", which means that for some fixed B, every H โˆˆโ„‹ 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 โ„‹, and (II) all tree pattern classes, i.e., all classes โ„‹ such that every Hโˆˆโ„‹ 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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