Finer-Grained Reductions in Fine-Grained Hardness of Approximation
Bibliographic record
Abstract
We investigate the relation between $δ$ and $ε$ required for obtaining a $(1+δ)$-approximation in time $N^{2-ε}$ for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity. Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension $c \log N$ in time $N^{2-ε}$, then there is no $(1+δ)$-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time $N^{2-2ε}$, where $δ\approx (ε/c)^2$ (where $\approx$ hides $\polylog$ factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of $δ$ on $ε$, on the order of $δ\approx (ε/c)^6$. Our result implies in turn that no $(1+δ)$-approximation algorithm exists for Euclidean closest pair for $δ\approx ε^4$, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of $δ\approx ε^3$ for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation. Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).
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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 | 0.030 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.005 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.017 |
| Open science | 0.006 | 0.009 |
| Research integrity | 0.003 | 0.010 |
| Insufficient payload (model declined to judge) | 0.017 | 0.003 |
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".