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, using 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 β-skeleton heuristic to certify that certain edges belong to the minimum-weight triangulation.

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Minimum-weight triangulation is NP-hard

that computes a subgraph of the minimum weight triangulation (A4WL”) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give two variants of our algorithm that produce a more complete subgraph of the MWT: an extended LMT-skeleton requiring worst case 0(n6) time and O(n3) space; and a modified LMT-skeleton requiring 0(n2) time and 0( TZ1”5) space in the expected case for uniform distributions.

A (usually?) connected subgraph of the minimum weight triangulation

This paper considers the problem of computing a minimum weight triangulation of n points in the plane, which has been intensively studied in recent years in computational geometry. This paper investigates a branch-and-cut approach for minimum weight triangulations. The problem can be formulated as finding a minimum-weight maximal independent set of intersection graphs of edges. In combinatorial optimization, there are known many cuts for the independent set problem, and we further use a cut induced by geometric properties of triangulations. Combining this branch-and-cut approach with the β-skeleton method, the moderate-size problem could be solved efficiently in our computational experiments. Polyhedral characterizations of the proposed cut and applications of another old skeletal approach in mathematical programming as the independent set problem are also touched upon.

A Branch-and-Cut Approach for Minimum Weight Triangulation

A triangulation of a given set S of n points in the plane is a maximal set of non-crossing line segments (called edges) which have both endpoints in S. A triangulation partitions the interior of the convex hull of the given point set into triangles. It is used in many areas of engineering and scientific applications such as finite element methods, approximation theory, numerical computation, computer-aided geometric design, and etc.

Jeffrey Cf Tsang

Papers

Expected case analysis of-skeletons with applications to the construction of minimum weight triangulations