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 ...

In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface that separates two sets. In this paper, we provide approximation algorithms for several geometric separation problems, including: • Given a set of triangles T and a set S of points that lie within the union of the triangles, find a minimum-cardinality set, T ′ , of pairwise-disjoint triangles, each contained within some triangle of T, that cover the point set S. • Given finite sets of “red ” and “blue ” points in the plane, determine a simple polygon of fewest edges that separates the red points from the blue points. More generally, given finite sets of points of many color classes, determine a planar “separating ” subdivision of minimum combinatorial complexity, which has the property that each face of the subdivision contains points of at most one color class; • Given two polyhedral terrains, P and Q, over a common support set (e.g., the unit square), with P lying above Q, compute a nested polyhedral terrain R that lies between P and Q such that R has a minimum number of facets. Exact solution of the above problems in polynomial time is highly unlikely: The decision versions of all three problems are known to be NP-hard. We provide polynomial-time algorithms that are guaranteed to produce an answer within a logarithmic factor (O(log n), where n is the complexity of the input problem instance) of optimal. (The error factor is constant in the orthogonal case — coverage by disjoint aligned rectangles, or separation of orthohedral terrains.) We also discuss extensions to higher dimensions. 1