Next: Problem 2: Voronoi Diagram of Moving Points
Can a minimum weight triangulation of a planar point set be found in polynomial time? The weight of a triangulation is its total edge length.
Perhaps E. L. Lloyd, 1977: [Llo77], cited in Garey and Johnson [GJ79].
Just solved by Wolfgang Mulzer and Günter Rote, January 2006! http://arxiv.org/abs/cs.CG/0601002. Entry to be updated later...
This problem is one of the few from Garey and Johnson [GJ79], p. 288 whose complexity status remains unknown.
The best approximation algorithms achieve a (large) constant times the optimal length [LK96]; good heuristics are known [DMM95]. If Steiner points are allowed, again constant-factor approximations are known [Epp94], [CL98], but it is open to decide even if a minimum-weight Steiner triangulation exists (the minimum might be approached only in the limit).
J. O’Rourke, 31 Jul. 2001; J. O’Rourke, 3 Jan. 2006.
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M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, NY, 1979.
Errol Lynn Lloyd. On triangulations of a set of points in the plane. In Proc. 18th Annu. IEEE Sympos. Found. Comput. Sci., pages 228–240, 1977.
Christos Levcopoulos and Drago Krznaric. Quasi-greedy triangulations approximating the minimum weight triangulation. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 392–401, 1996.
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