7 research outputs found
A generalization of the integer linear infeasibility problem
Does a given system of linear equations with nonnegative constraints have an
integer solution? This is a fundamental question in many areas. In statistics
this problem arises in data security problems for contingency table data and
also is closely related to non-squarefree elements of Markov bases for sampling
contingency tables with given marginals. To study a family of systems with no
integer solution, we focus on a commutative semigroup generated by a finite
subset of and its saturation. An element in the difference of the
semigroup and its saturation is called a ``hole''. We show the necessary and
sufficient conditions for the finiteness of the set of holes. Also we define
fundamental holes and saturation points of a commutative semigroup. Then, we
show the simultaneous finiteness of the set of holes, the set of non-saturation
points, and the set of generators for saturation points. We apply our results
to some three- and four-way contingency tables. Then we will discuss the time
complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5,
Issue 1 (2008) p36-5
Bounds on generalized Frobenius numbers
Let and let be relatively prime integers.
The Frobenius number of this -tuple is defined to be the largest positive
integer that has no representation as where
are non-negative integers. More generally, the -Frobenius
number is defined to be the largest positive integer that has precisely
distinct representations like this. We use techniques from the Geometry of
Numbers to give upper and lower bounds on the -Frobenius number for any
nonnegative integer .Comment: We include an appendix with an erratum and addendum to the published
version of this paper: two inaccuracies in the statement of Theorem 2.2 are
corrected and additional bounds on s-Frobenius numbers are derive
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