Minimum-weight triangulation is NP-hard

TL;DR

This paper proves that finding the minimum-weight triangulation of a planar point set is NP-hard by reducing from PLANAR-1-IN-3-SAT and using computer-assisted verification methods.

Contribution

It establishes the NP-hardness of the minimum-weight triangulation problem through a novel reduction and computational verification techniques.

Findings

NP-hardness of MWT proven via reduction from PLANAR-1-IN-3-SAT

Computer-assisted gadget verification using dynamic programming and beta-skeleton heuristic

Provides a foundation for understanding computational complexity of geometric triangulation problems

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.