Early on in the first project period, we managed to upgrade the "main engine" of the project: It was known and crucially exploited that, under certain technical conditions, linear combinations of homomorphism counts from fixed graphs are exactly as hard as their hardest terms. These technical conditions required the counts to come from small fixed patterns, and this seemed like an inherent requirement. We could however show that a similar statement holds even when the patterns are large. Our upgraded result implies new hardness results for counting patterns even when pattern and target graph have similar sizes. This enlarges the project scope.
Moreover, due to the results obtained so far, we now have a much better understanding of the complexity of a particular pattern counting problem, namely counting small induced subgraphs satisfying a fixed property: Given a large graph and a small number k, we would like to count all induced subgraphs with k vertices that satisfy some fixed property. For example, we might want to count all connected induced k-vertex subgraphs. Such problems have been studied for over a decade, and numerous techniques have been used to obtain hardness results that cover increasingly general (but still restricted) cases of the problem. We found that one single technique explains much of the complexity of this problem: Fourier analysis of Boolean functions. With this insight, we could prove hardness for a larger collection of properties and for variants of the problem. Most surprising to us was that all these results could be derived as relatively straightforward consequences from known theorems in Fourier analysis. Later, we also disproved the main conjecture in this area.
We also designed a new framework for proving complexity-theoretic lower bounds on homomorphism counts under the so-called exponential time hypothesis. Such lower bounds for homomorphism counts and related problems play a central role in parameterized complexity. Our new framework is built around a new structural graph parameter that is connected to routing problems in graphs, and by showing lower bounds on this graph parameter, we could directly obtain complexity-theoretic lower bounds. In particular, this technique allowed us to obtain new lower bounds and to prove known lower bounds in a more straightforward way.
Other results include novel insights into the distinguishing power of homomorphism counts and into combinatorial interpretations of linear combinations of homomorphism counts. We also extended connections between conjectures from linear algebra and algorithmic problems and found new such connections for so-called convolution problems. Moreover, we studied restrictions of the fundamental notion of treewidth and found links to the complexity of polynomials in depth-restricted computational models.
