Abstract
We study the fundamental problem of polytope membership aiming at convex polytopes in high dimension and with many facets, given as an intersection of halfspaces. Standard data-structures and brute force methods cannot scale, due to the curse of dimensionality. We design an efficient algorithm, by reduction to the approximate Nearest Neighbor (ANN) problem based on the construction of a Voronoi diagram with the polytope being one bounded cell. We thus trade exactness for efficiency so as to obtain complexity bounds polynomial in the dimension, by exploiting recent progress in the complexity of ANN search. We present a novel data structure for boundary queries based on a Newton-like iterative intersection procedure. We implement our algorithms and compare with brute-force approaches to show that they scale very well as the dimension and number of facets grow larger.
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Acknowledgements
The first two authors are partially supported by the European Union’s H2020 research and innovation programme under grant agreement No 734242.
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Anagnostopoulos, E., Emiris, I.Z., Fisikopoulos, V. (2018). Polytope Membership in High Dimension. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_4
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DOI: https://doi.org/10.1007/978-3-319-96151-4_4
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