Surface approximation and geometric partitions

Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise-linear function \Σ(x; y) of minimum complexity (that is, a xy-monotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that j\Σ(x p ; y p ) \Γ z p j "; for all (x p ; y p ; z p ) 2 S: We prove that the decision version of this problem is NP-Hard . The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise-linear surface of size O(K o log K o ), where K o is the complexity of an optimal surface satisfying the constraints of the problem. The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instan...