Go backward to 5 Face Lattice of Geometric Polytopes
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| Input: | Polytope P in H-description |
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| Output: | "Yes" if P not simple (degenerate), "No" otherwise |
| Status (general): | Strongly NP-complete |
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| Status (fixed dim.): | Polynomial time |
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Independently proved to be NP-complete in the papers of
Chandrasekaran, Kabadi, and Murty [11] and Dyer
[14]. Fukuda, Liebling, and Margot [22] proved
that the problem is strongly NP-complete. For fixed
dimension, one can enumerate all vertices in polynomial time (see
Problem 1) and check whether they are
simple or not. Bremner, Fukuda, and Marzetta [8] noted that if P is given in V-description the problem is polynomial time solvable: enumerate the edges (1-skeleton, see Problem 5) and apply the Lower Bound Theorem. Erickson [19] showed that in the worst case Ω(m⌈d/2⌉-1 + m log m) sideness queries are required to test whether a polytope is simple. For odd d this matches the upper bound. A sideness query is a question of the following kind: given d+1 points p0, …, pd in Rd, does p0 lie "above", "below", or on the oriented hyperplane determined by p1, p2, …, pd. |
| Related problems: | 1, 5 |
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