A quasi-polynomial time approximation scheme for minimum weight triangulation

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 NP-hard by Mulzer and Rote [2006]. In this article, we present a quasi-polynomial time approximation scheme for Minimum Weight Triangulation.

[1]  Jack Snoeyink,et al.  On computing edges that are in all minimum-weight triangulations , 1996, SCG '96.

[2]  David Eppstein.  Approximating the minimum weight steiner triangulation , 1994, Discret. Comput. Geom..

[3]  Franz Aurenhammer,et al.  On minimum weight pseudo-triangulations , 2009, Comput. Geom..

[4]  Errol L. Lloyd.  On triangulations of a set of points in the plane , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[5]  Joseph S. B. Mitchell,et al.  Computational Geometry Column 42 , 2001, Int. J. Comput. Geom. Appl..

[6]  Derek G. Corneil,et al.  Polynomial-time Instances of the Minimum Weight Triangulation Problem , 1993, Comput. Geom..

[7]  Christian Borgelt,et al.  Minimum Weight Triangulation by Cutting Out Triangles , 2005, ISAAC.

[8]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[9]  Randeep Bhatia,et al.  Book review: Approximation Algorithms for NP-hard Problems. Edited by Dorit S. Hochbaum (PWS, 1997) , 1998, SIGA.

[10]  Günter Rote,et al.  Minimum-weight triangulation is NP-hard , 2006, JACM.

[11]  G. Klincsek.  Minimal Triangulations of Polygonal Domains , 1980 .

[12]  StegerAngelika,et al.  A quasi-polynomial time approximation scheme for minimum weight triangulation , 2009 .

[13]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[14]  Sanjeev Arora,et al.  Approximation Schemes for Degree-Restricted MST and Red–Blue Separation Problems , 2004, Algorithmica.

[15]  Christos Levcopoulos,et al.  Quasi-greedy triangulations approximating the minimum weight triangulation , 1996, SODA '96.

[16]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[17]  Peter Gilbert.  New Results on Planar Triangulations. , 1979 .

[18]  Sanjeev Arora,et al.  Approximation schemes for NP-hard geometric optimization problems: a survey , 2003, Math. Program..

[19]  Tiow Seng Tan,et al.  A quadratic time algorithm for the minmax length triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[20]  Gautam Das,et al.  WHICH TRIANGULATIONS APPROXIMATE THE COMPLETE GRAPH? , 2022 .

[21]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[22]  D. Eppstein,et al.  Approximation algorithms for geometric problems , 1996 .

[23]  Michael Hoffmann,et al.  The minimum weight triangulation problem with few inner points , 2006, Comput. Geom..

[24]  Tiow Seng Tan,et al.  An O(n2 log n) Time Algorithm for the Minmax Angle Triangulation , 1992, SIAM J. Sci. Comput..

[25]  Joseph S. B. Mitchell,et al.  Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem , 1996, SODA '96.

[26]  Matthew Dickerson,et al.  A Large Subgraph of the Minimum Weight Triangulation , 1997, Discret. Comput. Geom..

[27]  Jack Snoeyink,et al.  On exclusion regions for optimal triangulations , 2001, Discret. Appl. Math..

[28]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[29]  Tiow Seng Tan,et al.  A Quadratic Time Algorithm for the Minimax Length Triangulation , 1993, SIAM J. Comput..

[30]  Satish Rao,et al.  Approximation schemes for Euclidean k-medians and related problems , 1998, STOC '98.

[31]  Yin-Feng Xu,et al.  Approaching the largest β-skeleton within a minimum weight triangulation , 1996, SCG '96.

[32]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[33]  Christos Levcopoulos,et al.  A near-optimal heuristic for minimum weight triangulation of convex polygons , 1997, SODA '97.

[34]  David A. Plaisted,et al.  A Heuristic Triangulation Algorithm , 1987, J. Algorithms.

[35]  Yin-Feng Xu,et al.  A Chain Decomposition Algorithm for the Proof of a Property on Minimum Weight Triangulations , 1994, ISAAC.

[36]  J. Mark Keil,et al.  Computing a Subgraph of the Minimum Weight Triangulation , 1994, Comput. Geom..