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

We propose to look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n 3) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6 k n 5log n) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k=O(log n). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k interior points.

The Minimum Weight Triangulation Problem with Few Inner Points

We look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n3) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6kn5 logn) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k = O(log n). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k inner points.

/pdf/the-minimum-weight-triangulation-problem-with-few-inner-45f07ureqp.pdf

The minimum weight triangulation problem with few inner points

Triangulation of a given set of points in a plane is one of the most commonly solved problems in computer graphics and computational geometry. Because they are useful in many applications, triangulations must provide well-shaped triangles. Many criteria have been developed to provide such meshes, namely weight and angular criteria. Each criterion has its pros and cons, some of them are difficult to compute, and sometimes even the polynomial algorithm is not known. By any of the existing deterministic methods, it is not possible to compute a triangulation which satisfies more than one criterion or which contains parts developed according to several criteria. We explain how such a mixture can be generated using genetic optimization.

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Multicriteria-optimized triangulations

We discuss and compare four fixed parameter algorithms for finding the minimum weight triangulation of a simple polygon with (n - k) vertices on the perimeter and k vertices in the interior (hole vertices), that is, for a total of n vertices. All four algorithms rely on the same abstract divide-and-conquer scheme, which is made efficient by a variant of dynamic programming. They are essentially based on two simple observations about triangulations, which give rise to triangle splits and paths splits. While each of the first two algorithms uses only one of these split types, the last two algorithms combine them in order to achieve certain improvements and thus to reduce the time complexity. By discussing this sequence of four algorithms we try to bring out the core ideas as clearly as possible and thus strive to achieve a deeper understanding as well as a simpler specification of these approaches. In addition, we implemented all four algorithms in Java and report results of experiments we carried out with this implementation. (Less)

https://www.borgelt.net/papers/tr_238_orig.pdf

Fixed parameter algorithms for the minimum weight triangulation problem

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.

Yoshiaki Kyoda

Papers

A Branch-and-Cut Approach for Minimum Weight Triangulation