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Minimum-weight triangulation is NP-hard

Wolfgang Mulzer·Guenter Rote
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Abstract

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.

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Wolfgang Mulzer, Guenter Rote. 2008-02-13. Minimum-weight triangulation is NP-hard. https://doi.org/10.1145/1346330.1346336

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