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The MINIMUM WEIGHT TRIANGULATION problem is to find a triangulation T * of minimum length for a given set of points P in the Euclidean plane. It was one of the few longstanding open problems from the famous list of twelve problems with unknown complexity status, published by Garey and Johnson [1979]. Very recently, the problem was shown to be N P-hard by Mulzer and Rote [2006]. In this article, we present a quasi-polynomial time approximation scheme for MINIMUM WEIGHT TRIANGULATION. ACM Reference Format:Remy, J., and Steger, A. 2009. A quasi-polynomial time approximation scheme for minimum weight triangulation.

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In this paper we introduce a new technique for approximation schemes for geometrical optimization problems. As an example problem, we consider the following variant of the geometric Steiner tree problem. Every point u which is not included in the tree costs a penalty of π(u) units. Furthermore, every Steiner point that we use costs c S units. The goal is to minimize the total length of the tree plus the penalties. Our technique yields a polynomial time approximation scheme for the problem, if the points lie in the plane.

This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniformly distributed over [0,1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves E[ALG]/E[OPT] = O(1) and E[ALG/OPT] = O(1) against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than E[ALG]/E[OPT] = Ω(log n) if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields E[ALG]/E[OPT] = O(log n) against the strongest-imaginable adversary.