The Salesman's Improved Tours for Fundamental Classes
Why this work is in the frame
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Bibliographic record
Abstract
Finding the exact integrality gap $\\alpha$ for the LP relaxation of the\nmetric Travelling Salesman Problem (TSP) has been an open problem for over\nthirty years, with little progress made. It is known that $4/3 \\leq \\alpha \\leq\n3/2$, and a famous conjecture states $\\alpha = 4/3$. For this problem,\nessentially two "fundamental" classes of instances have been proposed. This\nfundamental property means that in order to show that the integrality gap is at\nmost $\\rho$ for all instances of metric TSP, it is sufficient to show it only\nfor the instances in the fundamental class. However, despite the importance and\nthe simplicity of such classes, no apparent effort has been deployed for\nimproving the integrality gap bounds for them. In this paper we take a natural\nfirst step in this endeavour, and consider the $1/2$-integer points of one such\nclass. We successfully improve the upper bound for the integrality gap from\n$3/2$ to $10/7$ for a superclass of these points, as well as prove a lower\nbound of $4/3$ for the superclass. Our methods involve innovative applications\nof tools from combinatorial optimization which have the potential to be more\nbroadly applied.\n
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.006 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.010 | 0.006 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it