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 kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, 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 $P_A(b)=\{x: Ax=b, x\geq0\}$. 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 $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ 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 $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ 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 $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, 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 $(2,2)$ 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