Abstract
Since various measurements of convex polytopes play an important role in many applications, it is useful to know how efficiently such measurements can be computed or at least approximated. The present study, set in Euclidean n-space, focuses on the efficiency of polynomial-time algorithms for computing or approximating four “radii” of polytopes (diameter, width, inradius, circumradius) and the maximum of the Euclidean norm over a polytope. These functionals are known to be tractable in some cases, and the tractability results are here complemented by showing for each of the remaining cases that unless ℙ = ℕℙ, the performance ratios of polynomial-time approximation algorithms are uniformly bounded away from 1. These inapproximability results are established by means of a transformation from the problem Max-Not-All-Equal-3-Sat, and they apply even to very small classes of familiar polytopes (simplices, parallelotopes, and close relatives). They are sharp in the sense that the related problems are indeed approximable within a constant performance ratio. The results for parallelotopes apply also to the quadratic pseudoboolean optimization problems of maximizing a positive definite quadratic form over [0, l]n or [−1,1]n.
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Brieden, A., Gritzmann, P., Klee, V. (2000). Inapproximability of some Geometric and Quadratic Optimization Problems. In: Pardalos, P.M. (eds) Approximation and Complexity in Numerical Optimization. Nonconvex Optimization and Its Applications, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3145-3_7
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