Criticism of Hunting Minimum Weight Triangulation Edges

Minimum Weight Triangulation problem (MWT) is to nd a set of edges of minimum total length that triangulates a given set of points in the plane. Although some properties of MWT have been proved and many heuristics proposed, polyno-miality (and)/or NP-completness of MWT problem is still unsolved. The problem belongs to the few open problems from the book GaJo]. In this paper we present results indicating that even very good approaches based on the local edge examination (like LMT-skeleton) fail to come close to the MWT for specially constructed set of points. Furthermore, we show a method how to construct a set of points for which MWT is unstable | a slight displacement of a point in the input set causes signiicant change in the price of MWT. 1. Local MWT edge examination Up to now it is not known yet if there is a polynomial algorithm which nds the MWT for an arbitrary point set. Known polynomial algorithms related to MWT problem can be divided into two groups: 1. algorithms approximating the MWT by a triangulation which may not be the minimum one; 2. algorithms which attempt to nd a maximum subgraph of MWT. 1.1. Algorithms approximating the MWT. Algorithms belonging to the rst group are usually heuristics based on observing some features of the MWT. Here is a list of several such algorithms: Delaunay triangulation; greedy algorithm; Plaisted's and Hong's heuristics PlHo]; simulated annealing BFMNPS]; pairwise edge acceptation BFMNPS]; etc. However, all the algorithms mentioned above compute only an approximation of MWT (a special case is simulated annealing, which is a probabilistic algorithm).