Computing maxmin edge length triangulations
Bibliographic record
Abstract
In 1991, Edelsbrunner and Tan gave an $O(n^2)$ algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate MELT within any polynomial factor. While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of non-trivial size. A straightforward IP based on pairwise disjointness of the $\Theta(n^2)$ segments between the n points has $\Theta(n^4)$ constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small $n$. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to $n = 200$. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets.
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How this classification was reachedexpand
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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".