Neither the greedy nor the delaunay triangulation of a planar point set approximates the optimal triangulation

Consider the following heuristic for planar Euclidean instances of the traveling salesman problem (TSP): select a subset of the edges which induces a planar graph, and solve either the TSP or its graphical relaxation on that graph. In this paper, we give several motivations for considering this heuristic, along with extensive computational results. It turns out that the Delaunay and greedy triangulations make effective choices for the induced planar graph. Indeed, our experiments show that the resulting tours are on average within 0.1% of optimality.

The traveling salesman problem (TSP) is a fundamental and well-known problem in combinatorial optimisation. It has many applications, for example in vehicle routing and machine scheduling. This paper proposes several heuristics methods for the Euclidean TSP, based on the use of triangulations, and gives extensive computational results.

A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. We study the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but not Delaunay drawable – that is, not drawable as a Delaunay triangulation.

An exclusion region for a triangulation is a region that can be placed around each edge of the triangulation such that the region cannot contain points from the set on both sides of the edge. We survey known exclusion regions for several classes of triangulations, including Delaunay, Greedy, and Minimum Weight triangulations. We then show an exclusion region of larger area than was previously known for the minimum weight triangulation, which significantly speeds up an algorithm of Beirouti and Snoeyink. We also show that no exclusion region exists for the general class of locally optimal triangulations, in which every triangulation edge optimally triangulates the region determined by its two incident triangles.

This article settles the following two longstanding open problems:

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  What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation?

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  Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?

We present a new method for testing compatibility of candidate edges in the greedy triangulation, and new results on the rank of edges in various triangulations. Our edge test requires O(1) time for test and update, O(n) space, and O(n) time to initialize. Based on these results, we present fast greedy triangulation algorithms with expected case running time of O(n log n) for uniform distributions over convex regions. While algorithms with O(n) expected case running times exist, the algorithms presented here are simpler to implement and work well in practice.

We consider the problem of characterizing those graphs that can be drawn as minimum weight triangulations and answer the question for maximal outerplanar graphs. We provide a complete characterization of minimum weight triangulations of regular polygons by studying the combinatorial properties of their dual trees. We exploit this characterization to devise a linear time (real RAM) algorithm that receives as input a maximal outerplanar graph G and produces as output a straight-line drawing of G that is a minimum weight triangulation of the set of points representing the vertices of G.