Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron. Fukuda, K., Liebling, T. M., & Margot, F. *Computational Geometry*, 8(1):1--12, June, 1997. Paper doi abstract bibtex In this paper, we investigate the applicability of backtrack technique to solve the vertex enumeration problem and the face enumeration problem for a convex polyhedron given by a system of linear inequalities. We show that there is a linear-time backtrack algorithm for the face enumeration problem whose space complexity is polynomial in the input size, but the vertex enumeration problem requires a backtrack algorithm to solve a decision problem, called the restricted vertex problem, for each output, which is shown to be NP-complete. Some related NP-complete problems associated with a system of linear inequalities are also discussed, including the optimal vertex problems for polyhedra and arrangements of hyperplanes.

@article{ Fukuda1997,
abstract = {In this paper, we investigate the applicability of backtrack technique to solve the vertex enumeration problem and the face enumeration problem for a convex polyhedron given by a system of linear inequalities. We show that there is a linear-time backtrack algorithm for the face enumeration problem whose space complexity is polynomial in the input size, but the vertex enumeration problem requires a backtrack algorithm to solve a decision problem, called the restricted vertex problem, for each output, which is shown to be NP-complete. Some related NP-complete problems associated with a system of linear inequalities are also discussed, including the optimal vertex problems for polyhedra and arrangements of hyperplanes.},
author = {Fukuda, Komei and Liebling, Thomas M. and Margot, Fraņ{c}ois},
doi = {10.1016/0925-7721(95)00049-6},
file = {:Users/KunihiroWASA/Dropbox/paper/1997/Fukuda, Liebling, Margot, Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron, 1997.pdf:pdf},
issn = {09257721},
journal = {Computational Geometry},
month = {June},
number = {1},
pages = {1--12},
title = {{Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron}},
url = {http://www.sciencedirect.com/science/article/pii/0925772195000496 http://linkinghub.elsevier.com/retrieve/pii/0925772195000496},
volume = {8},
year = {1997}
}

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