720 research outputs found
A polyhedral Frobenius theorem with applications to integer optimization
We prove a representation theorem of projections of sets of integer points by an integer matrix . Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of nonlinear integer optimization problems in variable dimension. Concretely, we aim to optimize over a set , where is a nonlinear function, is a polyhedron, and . As a consequence of our representation theorem, we obtain a general efficient transformation from the latter class of problems to integer linear programming. Our bounds depend polynomially on various important parameters of the input data leading, among others, to first polynomial time algorithms for several classes of nonlinear optimization problems. Read More: http://epubs.siam.org/doi/10.1137/14M097369
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
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
Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations
We resolve the local semistable reduction problem for overconvergent
F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and
residue transcendence degree 0). We first introduce a higher-dimensional
analogue of the generic radius of convergence for a p-adic differential module,
which obeys a convexity property. We then combine this convexity property with
a form of the p-adic local monodromy theorem for so-called fake annuli.Comment: 36 pages; v3: refereed version; adds appendix with two example
Forall-exist statements in pseudopolynomial time
Given a convex set and an integer matrix , we consider statements of the form s.t. . Such statements can be verified in polynomial time
with the algorithm of Kannan and its improvements if is fixed and is a
polyhedron. The running time of the best-known algorithms is doubly exponential
in~. In this paper, we provide a pseudopolynomial-time algorithm if is
fixed. Its running time is , where . Furthermore it applies to general convex sets . Second, we
provide new upper bounds on the \emph{diagonal} as well as the \emph{polyhedral
Frobenius} number, two recently studied forall-exist problems
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
Presburger arithmetic, rational generating functions, and quasi-polynomials
Presburger arithmetic is the first-order theory of the natural numbers with
addition (but no multiplication). We characterize sets that can be defined by a
Presburger formula as exactly the sets whose characteristic functions can be
represented by rational generating functions; a geometric characterization of
such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 page
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