4 research outputs found
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
Complexity of short Presburger arithmetic
We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short" we mean sentences with a bounded number of variables,
quantifiers, inequalities and Boolean operations; the input consists only of
the integers involved in the inequalities. We prove that assuming Kannan's
partition can be found in polynomial time, the satisfiability of Short-PA
sentences can be decided in polynomial time. Furthermore, under the same
assumption, we show that the numbers of satisfying assignments of short
Presburger sentences can also be computed in polynomial time
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives