Bibliographic record
Abstract
For any fixed graph $G$, the subgraph isomorphism problem asks whether an $n$-vertex input graph has a subgraph isomorphic to $G$. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted $G$-$\mathsf{SUB}$, and then solves $G$-$\mathsf{SUB}$ in time $O(n^{tw(G)+1})$ where $tw(G)$ is the treewidth of $G$. Marx (2010) conjectured that $G$-$\mathsf{SUB}$ requires time $Ω(n^{\mathrm{const}\cdot tw(G)})$ and, assuming the Exponential Time Hypothesis, proved a lower bound of $Ω(n^{\mathrm{const}\cdot emb(G)})$ for a certain graph parameter $emb(G) \ge Ω(tw(G)/\log tw(G))$. With respect to the size of $\mathrm{AC}^0$ circuits solving $G$-$\mathsf{SUB}$ in the average case, Li, Razborov and Rossman (2017) proved (unconditional) upper and lower bounds of $O(n^{2κ(G)+\mathrm{const}})$ and $Ω(n^{κ(G)})$ for a different graph parameter $κ(G) \ge Ω(tw(G)/\log tw(G))$. Our contributions are as follows. First, we prove that $emb(G)$ is $O(κ(G))$ for all graphs $G$. Next, we show that $κ(G)$ can be asymptotically less than $tw(G)$; for example, if $G$ is a hypercube then $κ(G)$ is $Θ\big(tw(G)\big/\sqrt{\log tw(G)}\big)$. This implies that the average-case complexity of $G$-$\mathsf{SUB}$ is $n^{o(tw(G))}$ when $G$ is a hypercube. Finally, we construct $\mathrm{AC}^0$ circuits of size $O(n^{κ(G)+\mathrm{const}})$ that solve $G$-$\mathsf{SUB}$ in the average case, closing the gap between the upper and lower bounds of Li et al.
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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 | 0.005 | 0.048 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.004 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.007 | 0.026 |
| Open science | 0.008 | 0.009 |
| Research integrity | 0.005 | 0.011 |
| Insufficient payload (model declined to judge) | 0.025 | 0.007 |
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".