Abstract.
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard.
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Received: June 21, 2001 Final version received: July 29, 2002
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ID="*" Supported by a DFG Gerhard-Hess-Forschungsförderungspreis donated to Günter M. Ziegler (Zi 475/2-3)
MSC 2000: 52B05, 05C60, 52B11, 68R10
Acknowledgments. We are indebted to Günter M. Ziegler for careful reading an earlier version of the paper and for stimulating discussions, in particular, for bringing Bayer's work [3] to our attention. We thank Günter Rote for inspiring comments, Christian Knauer for pointing to Akutsu's results [1], and Christoph Eyrich for valuable LATEX advice.
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Kaibel, V., Schwartz, A. On the Complexity of Polytope Isomorphism Problems. Graphs and Combinatorics 19, 215–230 (2003). https://doi.org/10.1007/s00373-002-0503-y
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DOI: https://doi.org/10.1007/s00373-002-0503-y