Periodic Reporting for period 1 - CountHom (Counting (with) homomorphisms)
Okres sprawozdawczy: 2023-04-01 do 2025-09-30
This project addresses the computational complexity of counting problems in graphs; such problems ask to compute the number of certain structures rather than merely deciding their existence. Counting problems find applications in diverse areas like network analysis, machine learning, probabilistic databases, and statistical physics. They are linked to fundamental questions in complexity theory and give rise to algorithmic breakthroughs for decision problems.
This project will go beyond the state of the art in computational counting by building bridges between algorithms/complexity and the mathematical theory of graph homomorphisms, which are structure-preserving maps between graphs. Already starting from the 1960s, Lovász and others showed that homomorphism numbers between graphs tie together disparate mathematical areas. Similarly, the PI observed in the past 5 years that the computational problem of counting homomorphisms unifies a range of problems in computer science that were previously studied in isolation.
This connection allows us to approach fundamental questions in parameterized and fine-grained complexity related to the complexity of counting small patterns in large graphs, which were out of reach for more combinatorial approaches. Secondly, it allows us to revisit problems in classical counting complexity from a new angle, especially for partition functions in discrete physical systems, with the aim of simplifying, unifying and extending known results. Thirdly, homomorphism counts give a surprising novel viewpoint on questions in algebraic complexity surrounding the "permanent versus determinant" problem, an algebraic variant of the "P versus NP" problem.
This project will go beyond the state of the art in computational counting by building bridges between algorithms/complexity and the mathematical theory of graph homomorphisms, which are structure-preserving maps between graphs. Already starting from the 1960s, Lovász and others showed that homomorphism numbers between graphs tie together disparate mathematical areas. Similarly, the PI observed in the past 5 years that the computational problem of counting homomorphisms unifies a range of problems in computer science that were previously studied in isolation.
This connection allows us to approach fundamental questions in parameterized and fine-grained complexity related to the complexity of counting small patterns in large graphs, which were out of reach for more combinatorial approaches. Secondly, it allows us to revisit problems in classical counting complexity from a new angle, especially for partition functions in discrete physical systems, with the aim of simplifying, unifying and extending known results. Thirdly, homomorphism counts give a surprising novel viewpoint on questions in algebraic complexity surrounding the "permanent versus determinant" problem, an algebraic variant of the "P versus NP" problem.
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.
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.
Challenging follow-up questions emerged during the first phase of the project. For example, while we obtained somewhat stronger connections between homomorphism counts and general counting problems, it is still very nontrivial to apply these insights in the full generality envisioned. For this, we need to obtain an even better understanding of the mathematical theory of homomorphism counts, which we strive to obtain in the upcoming project period. This will allow us to apply the methodology of this project to more general graph parameters beyond small pattern counts.
Moreover, while we have identified new structural properties of graphs that yield lower bounds for counting problems under complexity-theoretic assumptions, we could also show that even our new techniques have some limitations that prevent them from pinpointing the exact complexity of several interesting counting problems. This means that new techniques are required for these bottleneck problems.
In the first project period, we also showed that a conjectured complexity classification for induced subgraph counts is in fact not true, as we could find counterexamples to the original conjecture. This will require us to formulate (and possibly prove) a new dichotomy conjecture.
Moreover, we have identified connections between our work and a conjecture from mathematics on the rank of certain tensors. A full resolution of this mathematical conjecture will likely not be achieved in this project, as it is tied to longstanding open problems. However, even partial progress on this conjecture will allow us to obtain improved algorithms, and it is also conceivable that algorithmic insights may also have implications on this conjecture.
Moreover, while we have identified new structural properties of graphs that yield lower bounds for counting problems under complexity-theoretic assumptions, we could also show that even our new techniques have some limitations that prevent them from pinpointing the exact complexity of several interesting counting problems. This means that new techniques are required for these bottleneck problems.
In the first project period, we also showed that a conjectured complexity classification for induced subgraph counts is in fact not true, as we could find counterexamples to the original conjecture. This will require us to formulate (and possibly prove) a new dichotomy conjecture.
Moreover, we have identified connections between our work and a conjecture from mathematics on the rank of certain tensors. A full resolution of this mathematical conjecture will likely not be achieved in this project, as it is tied to longstanding open problems. However, even partial progress on this conjecture will allow us to obtain improved algorithms, and it is also conceivable that algorithmic insights may also have implications on this conjecture.