New Exact Methods for Solving Quadratic Traveling Salesman Problem
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Bibliographic record
Abstract
The Quadratic Traveling Salesman Problem (QTSP) is a generalization of the Traveling Salesman Problem (TSP) with important applications in robotics and bioinformatics. The QTSP objective value depends on pairs of consecutive edges in the tour; hence, it is quadratic and generally hard to optimize. While various exact-solving approaches have been explored, many rely on specialized procedures and struggle to scale on large instances. More recently, carefully crafted metaheuristics have demonstrated better primal bounds and scalability, but they cannot provide any guarantees of solution quality nor prove the optimality of any solution. In this work, we propose new exact models for QTSP. We define direct encodings of QTSP in domain-independent dynamic programming (DIDP), constraint programming (CP), mixed integer quadratic programming (MIQP), and mixed integer linear programming (MILP), and compare them with the best-known exact method, a branch and cut (B&C) algorithm, and the state-of-the-art metaheuristic, a hybrid genetic algorithm (HGA). Our experimental results demonstrate that the DIDP model shows better scalability and finds the best feasible solutions on average among all exact solvers, including the B&C algorithm. HGA finds the best feasible solution among all approaches, with DIDP within 15% of the HGA cost on all experimented instances. Also, interestingly, our MILP model with the subtour elimination constraints generally finds better feasible solutions than the B&C algorithm while matching it in proving optimality, suggesting that lazily adding sub-tour elimination cuts is not particularly helpful in QTSP.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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