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On Simple Polygonalizations with Optimal Area

Abstract.

We discuss the problem of finding a simple polygonalization for a given set of vertices P that has optimal area. We show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P and prove that it is NP-complete to find a minimum weight polygon or a maximum weight polygon for a given vertex set, resulting in a proof of NP-completeness for the corresponding area optimization problems. This answers a generalization of a question stated by Suri in 1989. Finally, we turn to higher dimensions, where we prove that, for 1 \(\leq\) k d , 2 \(\leq\) d , it is NP-hard to determine the smallest possible total volume of the k -dimensional faces of a d -dimensional simple nondegenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980.

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Received June 26, 1997, and in revised form February 13, 1999, and May 19, 1999.

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Fekete, S. On Simple Polygonalizations with Optimal Area . Discrete Comput Geom 23, 73–110 (2000). https://doi.org/10.1007/PL00009492

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  • DOI: https://doi.org/10.1007/PL00009492

Keywords

  • High Dimension
  • Maximum Weight
  • Minimum Weight
  • Simple Polygon
  • Optimal Area