Abstract
We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and #mathsf {W[1]}-complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.
Highlights
In his seminal work on the complexity of computing the permanent, Valiant [56] introduced counting complexity which has since evolved into a well-studied sub-An extended abstract of this work has been published in the conference proceedings of the 25th Annual European Symposium on Algorithms (ESA 2017) [50].Algorithmica (2021) 83:1829–1860 field of computational complexity theory
The proof of the previous theorem relies on the following #P-hardness result regarding the subgraph counting problem restricted to trees; it is shown in Sect. 4.1 and might be of independent interest
For #P-hardness of the subgraph counting problem restricted to trees, we adapt the idea of the “skeleton graph” by Goldberg and Jerrum [28] and reduce directly from computing the permanent, the latter of which is #P-hard as shown by Valiant [56]
Summary
In his seminal work on the complexity of computing the permanent, Valiant [56] introduced counting complexity which has since evolved into a well-studied sub-. Examples of dichotomies in parameterized counting complexity are the complete classifications of the homomorphism counting problem due to Dalmau and Jonsson [19] and the subgraph counting problem due to Curticapean and Marx [17] For the latter, one is given graphs H and G and aims to count the number of subgraphs of G isomorphic to H , parameterized by the size of H. One only has to show two properties of (1) to obtain the dichotomy for counting subgraph embeddings: First, one has to show that a high matching number of H implies that one of the graphs H /ρ has high treewidth and second, that two (or more) terms with high treewidth and isomorphic graphs H /ρ and H /σ do not cancel out; note that the Möbius function might be negative. As there is a closed form for the Möbius function over the partition lattice it was possible to show that the sign of the Möbius function is equal whenever H /ρ and H /σ are isomorphic
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