An Approximation Algorithm for the Bipartite Traveling Tournament Problem
Bibliographic record
Abstract
The bipartite traveling tournament problem (BTTP) is an NP-complete scheduling problem whose solution is a double round-robin inter-league tournament with minimum total travel distance. The 2n-team BTTP is a variant of the well-known traveling salesman problem (TSP), albeit much harder as it involves the simultaneous coordination of 2n teams playing a sequence of home and away games under fixed constraints, rather than a single entity passing through the locations corresponding to the teams' home venues. As the BTTP requires a distance-optimal schedule linking venues in close proximity, we provide an approximation algorithm for the BTTP based on an approximate solution to the corresponding TSP. We prove that our polynomial-time algorithm generates a 2n-team inter-league tournament schedule whose total distance is at most 1 + 2c/3 + (3 − c)/(3n) times the total distance of the optimal BTTP solution, where c is the approximation factor of the TSP. In practice, the actual approximation factor is far better; we provide a specific example by generating a nearly-optimal inter-league tournament for the 30-team National Basketball Association, with total travel distance just 1.06 times the trivial theoretical lower bound.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".