Counting Homomorphisms Modulo a Prime Number
Bibliographic record
Abstract
Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) - the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_{p}GraphHom(H) problem, p a prime number: Given a graph G, find the number of homomorphisms from G to H modulo p. In a series of papers Faben and Jerrum, and Göbel et al. determined the complexity of #_{2}GraphHom(H) in the case H (or, in fact, a certain graph derived from H) is square-free, that is, does not contain a 4-cycle. Also, Göbel et al. found the complexity of #_{p}GraphHom(H) when H is a tree for an arbitrary prime p. Here we extend the above result to show that the #_{p}GraphHom(H) problem is #_{p}P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_{p}GraphHom(H) for square-free graphs H.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.027 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.022 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".