97 research outputs found
Computing the Face Lattice of a Polytope from its Vertex-Facet Incidences
We give an algorithm that constructs the Hasse diagram of the face lattice of
a convex polytope P from its vertex-facet incidences in time O(min{n,m}*a*f),
where n is the number of vertices, m is the number of facets, a is the number
of vertex-facet incidences, and f is the total number of faces of P. This
improves results of Fukuda and Rosta (1994), who described an algorithm for
enumerating all faces of a d-polytope in O(min{n,m}*d*f^2) steps. For simple or
simplicial d-polytopes our algorithm can be specialized to run in time
O(d*a*f). Furthermore, applications of the algorithm to other atomic lattices
are discussed, e.g., to face lattices of oriented matroids.Comment: 14 pages; to appear in: Comput. Geom.; the new version contains some
minor extensions and corrections as well as a more detailed treatment of
oriented matroid
On the Complexity of Polytope Isomorphism Problems
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.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
Computing the bounded subcomplex of an unbounded polyhedron
We study efficient combinatorial algorithms to produce the Hasse diagram of
the poset of bounded faces of an unbounded polyhedron, given vertex-facet
incidences. We also discuss the special case of simple polyhedra and present
computational results.Comment: 16 page
Reconstructing a Simple Polytope from its Graph
Blind and Mani (1987) proved that the entire combinatorial structure (the
vertex-facet incidences) of a simple convex polytope is determined by its
abstract graph. Their proof is not constructive. Kalai (1988) found a short,
elegant, and algorithmic proof of that result. However, his algorithm has
always exponential running time. We show that the problem to reconstruct the
vertex-facet incidences of a simple polytope P from its graph can be formulated
as a combinatorial optimization problem that is strongly dual to the problem of
finding an abstract objective function on P (i.e., a shelling order of the
facets of the dual polytope of P). Thereby, we derive polynomial certificates
for both the vertex-facet incidences as well as for the abstract objective
functions in terms of the graph of P. The paper is a variation on joint work
with Michael Joswig and Friederike Koerner (2001).Comment: 14 page
A face iterator for polyhedra and more general finite locally branched lattices
We discuss a new memory-efficient depth-first algorithm and its
implementation that iterates over all elements of a finite locally branched
lattices. This algorithm can be applied to face lattices of polyhedra and
various generalizations such as finite polyhedral complexes and subdivisions of
manifolds, extended tight spans and closed sets of matroids. Its practical
implementation is very fast compared to state-of-the-art implementations of
previously considered algorithms.Comment: 13 pages including long examples and computational dat
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
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