Minimum-weight triangulation is NP-hard
<|reference_start|>Minimum-weight triangulation is NP-hard: A triangulation of a planar point set S is a maximal plane straight-line graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths. We prove that the decision version of this problem is NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the gadgets is established with computer assistance, using dynamic programming on polygonal faces, as well as the beta-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.<|reference_end|>
@article{mulzer2006minimum-weight,
    title={Minimum-weight triangulation is NP-hard},
    author={Wolfgang Mulzer and Guenter Rote},
    journal={Journal of the ACM, 55, no. 2 (May 2008), Article 11, 29 pp.},
    year={2006},
    doi={10.1145/1346330.1346336},
    number={B 05-23 (revised)},
    archivePrefix={arXiv},
    eprint={cs/0601002},
    primaryClass={cs.CG cs.CC}
}