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On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
Abstract

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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected 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 computeraided 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 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...
Animal Testing
"... Abstract. A configuration of unit cubes in three dimensions with integer coordinates is called ananimaliftheboundaryoftheirunionis homeomorphic to a sphere. Shermer discovered several animals from which no single cube may be removed such that the resulting configurations are also animals [16]. Here ..."
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Abstract. A configuration of unit cubes in three dimensions with integer coordinates is called ananimaliftheboundaryoftheirunionis homeomorphic to a sphere. Shermer discovered several animals from which no single cube may be removed such that the resulting configurations are also animals [16]. Here we obtain a dual result: we give an example of an animal to which no cube may be added within its minimal bounding box such that the resulting configuration is also an animal. We also present two O(n)time algorithms for determining whether a given configuration of n unit cubes is an animal. 1