960 research outputs found
Polymake and Lattice Polytopes
The polymake software system deals with convex polytopes and related objects
from geometric combinatorics. This note reports on a new implementation of a
subclass for lattice polytopes. The features displayed are enabled by recent
changes to the polymake core, which will be discussed briefly.Comment: 12 pages, 1 figur
Minimizing the number of lattice points in a translated polygon
The parametric lattice-point counting problem is as follows: Given an integer
matrix , compute an explicit formula parameterized by that determines the number of integer points in the polyhedron . In the last decade, this counting problem has received
considerable attention in the literature. Several variants of Barvinok's
algorithm have been shown to solve this problem in polynomial time if the
number of columns of is fixed.
Central to our investigation is the following question: Can one also
efficiently determine a parameter such that the number of integer points in
is minimized? Here, the parameter can be chosen
from a given polyhedron .
Our main result is a proof that finding such a minimizing parameter is
-hard, even in dimension 2 and even if the parametrization reflects a
translation of a 2-dimensional convex polygon. This result is established via a
relationship of this problem to arithmetic progressions and simultaneous
Diophantine approximation.
On the positive side we show that in dimension 2 there exists a polynomial
time algorithm for each fixed that either determines a minimizing
translation or asserts that any translation contains at most times
the minimal number of lattice points
Algebraic Unimodular Counting
We study algebraic algorithms for expressing the number of non-negative
integer solutions to a unimodular system of linear equations as a function of
the right hand side. Our methods include Todd classes of toric varieties via
Gr\"obner bases, and rational generating functions as in Barvinok's algorithm.
We report polyhedral and computational results for two special cases: counting
contingency tables and Kostant's partition function.Comment: 21 page
Symbolic and analytic techniques for resource analysis of Java bytecode
Recent work in resource analysis has translated the idea of amortised resource analysis to imperative languages using a program logic that allows mixing of assertions about heap shapes, in the tradition of separation logic, and assertions about consumable resources. Separately, polyhedral methods have been used to calculate bounds on numbers of iterations in loop-based programs. We are attempting to combine these ideas to deal with Java programs involving both data structures and loops, focusing on the bytecode level rather than on source code
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
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