On the Complexity of Polytope Isomorphism Problems

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. MSC 2000: 52B05, 05C60, 52B11, 68R10 ⋆ Supported by a DFG Gerhard-Hess-Forschungsförderungspreis donated to Günter M. Ziegler (Zi 475/2-3).

Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation

“…Our classification problem is related to the isomorphism of polynomials problem [24] in multivariate cryptology [5], [10]. Support sets of polynomials span Newton polytopes and a related problems is the polytope isomorphism problem, see [18] for its complexity, which is as hard as the graph isomorphism problem.…”

Combined InSAR and Terrestrial Structural Monitoring of Bridges

“…Specifically, it is possible to understand the relationship between these measurements by taking into account the volume of the multidimensional subspace spanned by them. Indeed, it is worth recalling that a set of R records characterised by N attributes can be identified as a N -dimensional polytope [33]. According to a vertex geometry description of multidimensional datasets, every point living within the volume spanned by the R records in this subspace can be characterised as a proper function of these N -dimensional samples [34], [35].…”

“…. Given the nature of the elements in A and S, it is possible to prove that V A and V S are positive [33]. Furthermore, the aforementioned polytopes will cover a subspace in the N -dimensional space along the same directions [33]- [36].…”

Algorithms for highly symmetric linear and integer programs

“…Since x is integral coordinates of types (6) and (7) are integers for arbitrary parameters λ ∈ (0, 1). The coordinates of types (8) and (9)…”

“…The combinatorial automorphisms of a polytope are the (labeled) graph automorphisms of the bipartite graph encoded by the vertex-edge-incidences. This directly follows from the fact that the face lattice of a polytope is atomic and coatomic; see Kaibel and Schwartz [9]. Liberti studies automorphisms of optimization problems which are more general than integer linear programs [12].…”