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 $\leq$ kd , 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.

[1]  Eugene L. Lawler,et al.  Traveling Salesman Problem , 2016 .

[2]  J. Reeve.  On the Volume of Lattice Polyhedra , 1957 .

[3]  Leonidas J. Guibas,et al.  Finding extremal polygons , 1982, STOC '82.

[4]  David P. Dobkin,et al.  Finding Extremal Polygons , 1985, SIAM J. Comput..

[5]  Christos H. Papadimitriou,et al.  On Two Geometric Problems Related to the Traveling Salesman Problem , 1984, J. Algorithms.

[6]  I. Niven,et al.  Lattice Points and Polygonal Area , 1967 .

[7]  H. Coxeter,et al.  Introduction to Geometry. , 1961 .

[8]  Joseph S. B. Mitchell,et al.  Separation and approximation of polyhedral objects , 1992, SODA '92.

[9]  Joseph S. B. Mitchell,et al.  Approximation algorithms for geometric tour and network design problems (extended abstract) , 1995, SCG '95.

[10]  Ján Plesník,et al.  The NP-Completeness of the Hamiltonian Cycle Problem in Planar Digraphs with Degree Bound Two , 1979, Inf. Process. Lett..

[11]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[12]  David Eppstein,et al.  Finding minimum areak-gons , 1992, Discret. Comput. Geom..

[13]  Robert E. Tarjan,et al.  Rectilinear Planar Layouts and Bipolar Orientations of , 1986 .

[14]  Subhash Suri,et al.  Surface approximation and geometric partitions , 1994, SODA '94.

[15]  John R. Reay,et al.  The boundary characteristic and Pick's theorem in the Archimedean planar tilings , 1987, J. Comb. Theory, Ser. A.

[16]  Jeffrey S. Salowe.  Selecting the kth Largest-Area Convex Polygon , 1989, WADS.

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

[18]  Ann K. Stehney,et al.  Selected papers on geometry , 1983 .

[19]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[20]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[21]  Jayme Luiz Szwarcfiter,et al.  Hamilton Paths in Grid Graphs , 1982, SIAM J. Comput..