On the Complexity of Polytope Isomorphism Problems

Details On the Complexity of Polytope Isomorphism Problems On the Complexity of Polytope Isomorphism Problems

2001-06-12

arxiv.org/abs/math/0106093v2

Mathematics

Combinatorics

16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the Complexity of Isomorphism Problems Related to Polytopes'

Scientific paper

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. The original version of the paper (June 2001, 11 pages) had the title ``On the Complexity of Isomorphism Problems Related to Polytopes''. The main difference between the current and the former version is a new polynomial time algorithm for polytope isomorphism in bounded dimension that does not rely on Luks polynomial time algorithm for checking two graphs of bounded valence for isomorphism. Furthermore, the treatment of geometric isomorphism problems was extended.

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