78 research outputs found
Integer Polynomial Optimization in Fixed Dimension
We classify, according to their computational complexity, integer
optimization problems whose constraints and objective functions are polynomials
with integer coefficients and the number of variables is fixed. For the
optimization of an integer polynomial over the lattice points of a convex
polytope, we show an algorithm to compute lower and upper bounds for the
optimal value. For polynomials that are non-negative over the polytope, these
sequences of bounds lead to a fully polynomial-time approximation scheme for
the optimization problem.Comment: In this revised version we include a stronger complexity bound on our
algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time
approximation scheme) to maximize a non-negative integer polynomial over the
lattice points of a polytop
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
Notes about the Caratheodory number
In this paper we give sufficient conditions for a compactum in
to have Carath\'{e}odory number less than , generalizing an old result of
Fenchel. Then we prove the corresponding versions of the colorful
Carath\'{e}odory theorem and give a Tverberg type theorem for families of
convex compacta
Sparsity of integer solutions in the average case
We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Carathéodory number provided that the determinants in the data are small
Continuum Surface Energy from a Lattice Model
We investigate connections between the continuum and atomistic descriptions
of deformable crystals, using certain interesting results from number theory.
The energy of a deformed crystal is calculated in the context of a lattice
model with general binary interactions in two dimensions. A new bond counting
approach is used, which reduces the problem to the lattice point problem of
number theory. The main contribution is an explicit formula for the surface
energy density as a function of the deformation gradient and boundary normal.
The result is valid for a large class of domains, including faceted (polygonal)
shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints
corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26
pages. Abstract changed. Section 2 split into 2. Section (4) added material.
V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs
added.V 5,intro changed V.6 address reviewer's comment
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
On some algebraic identities and the exterior product of double forms
We use the exterior product of double forms to reformulate celebrated
classical results of linear algebra about matrices and bilinear forms namely
the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton
identities and Jacobi's formula for the determinant. This new formalism is then
used to naturally generalize the previous results to higher multilinear forms
namely to double forms.
In particular, we show that the Cayley-Hamilton theorem once applied to the
second fundamental form of a hypersurface of the Euclidean space is equivalent
to a linearized version of the Gauss-Bonnet theorem, and once its
generalization is applied to the Riemann curvature tensor (seen as a
double form) is an infinitisimal version of the general Gauss-Bonnet-Chern
theorem. In addition to that, the general Cayley-Hamilton theorems generate
several universal curvature identities. The generalization of the classical
Laplace expansion of the determinant to double forms is shown to lead to new
general Avez type formulas for all Gauss-Bonnet curvatures.Comment: 32 pages, in this new version we added: an introduction to the
exterior and composition products of double forms, a new section about
hyperdeterminants and hyperpfaffians and reference
An exact duality theory for semidefinite programming based on sums of squares
Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry: A linear matrix inequality is infeasible if and only
if -1 lies in the quadratic module associated to it. We also present a new
exact duality theory for semidefinite programming, motivated by the real
radical and sums of squares certificates from real algebraic geometry.Comment: arXiv admin note: substantial text overlap with arXiv:1108.593
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
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