The Two-Stripe Symmetric Circulant TSP is in P
Why this work is in the frame
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Bibliographic record
Abstract
The symmetric circulant TSP is a special case of the traveling salesman problem in which edge costs are symmetric and obey circulant symmetry. Despite the substantial symmetry of the input, remarkably little is known about the symmetric circulant TSP, and the complexity of the problem has been an often-cited open question. Considerable effort has been made to understand the case in which only edges of two lengths are allowed to have finite cost: the two-stripe symmetric circulant TSP. In this paper, we resolve the complexity of the two-stripe symmetric circulant TSP. To do so, we reduce two-stripe symmetric circulant TSP to the problem of finding certain minimum-cost Hamiltonian paths on cylindrical graphs. We then solve this Hamiltonian path problem. Our results show that the two-stripe symmetric circulant TSP is in P. Note that a two-stripe symmetric circulant TSP instance consists of a constant number of inputs (including $n$, the number of cities), so that a polynomial-time algorithm for the decision problem must run in time polylogarithmic in $n$, and a polynomial-time algorithm for the optimization problem cannot output the tour. We address this latter difficulty by showing that the optimal tour must fall into one of two parameterized classes of tours, and that we can output the class and the parameters in polynomial time. Thus we make a substantial contribution to the set of polynomial-time solvable special cases of the TSP, and take an important step towards resolving the complexity of the general symmetric circulant TSP.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 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