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How efficiently can one compute a polyhedral surface that is an \epsilon-approximation of a given triangulated surface in \R^3?
Mitchell [?]
Open.
It is NP-hard to obtain the minimum-facet surface separating two nested convex polytopes [DG97], if they are allowed to have distinct structures; the complexity is open when one polytope is an offset of the other (e.g., if they arise as offsets of an underlying convex polytope that we seek to approximate). Polynomial-time approximation algorithms are known ([BG95], [MS95], [AS98]) for the case of nested convex polyhedra, and for separating two terrain surfaces, achieving factors within O(1) or O(\log n) of optimal. However, no polynomial-time approximation results are known for general surfaces.
simplification
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