The traveling salesman problem for lines, balls and planes
Bibliographic record
Abstract
We revisit the traveling salesman problem with neighborhoods (TSPN) and obtain several approximation algorithms. These constitute either improvements over previously best approximations achievable in comparable times (for unit disks in the plane), or first approximations ever (for planes, lines and unit balls in 3-space).(I) Given a set of n planes in 3-space, a TSP tour that is at most 2.31 times longer than the optimal can be computed in O(n) time.(II) Given a set of n lines in 3-space, a TSP tour that is at most O(log3n) times longer than the optimal can be computed in polynomial time.(III) Given a set of n unit disks in the plane (resp., unit balls in 3-space), we improve the approximation ratio using a black box that computes an approximate tour for a set of points (the centers of a subset of the disks or the balls).
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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 | 0.000 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".