## On the Complexity of Optimization Problems for 3-Dimensional Convex Polyhedra and Decision Trees (1995)

Venue: | Comput. Geom. Theory Appl |

Citations: | 14 - 0 self |

### BibTeX

@ARTICLE{Das95onthe,

author = {Gautam Das and Michael T. Goodrich},

title = {On the Complexity of Optimization Problems for 3-Dimensional Convex Polyhedra and Decision Trees},

journal = {Comput. Geom. Theory Appl},

year = {1995},

volume = {8},

pages = {8--123}

}

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### Abstract

We show that several well-known optimization problems involving 3-dimensional convex polyhedra and decision trees are NP-hard or NP-complete. One of the techniques we employ is a linear-time method for realizing a planar 3-connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computer-aided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3-connected. Unfortunately, algorithmic problems dealing with 3-dimensional convex polyhedra ...