Lattice based extended formulations for integer linear equality systems (original) (raw)

A generalization of the integer linear infeasibility problem

Does a given system of linear equations Ax = b have a nonnegative integer solution? This is a fundamental question in many areas, such as operations research, number theory, and statistics. In terms of optimization, this is called an integer feasibility problem. A generalized integer feasibility problem is to find b such that there does not exist a nonnegative integral solution in the system with a given A. One such problem is the well-known Frobenius problem. In this paper we study the generalized integer feasibility problem and also the multi-dimensional Frobenius problem. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite subset of Z d 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. As examples we consider some three-and fourway contingency tables from statistics and apply our results to them. Then we will discuss the time complexities of our algorithms.

Integer Programming, Lattices, and Results in Fixed Dimension

Handbooks in Operations Research and Management Science, 2005

We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixed.

Propagating systems of dense linear integer constraints

2008

In interval propagation approaches to solving nonlinear constraints over reals it is common to build stronger propagators from systems of linear equations. This, as far as we are aware, is not pursued for integer finite domain propagation. In this paper we show how we use interval Gauss-Jordan elimination to build stronger propagators for an integer propagation solver. In a similar fashion we present an interval Fourier elimination preconditioning technique to generate redundant linear constraints from a system of linear inequalities. We show how to convert the resulting interval propagators into integer propagators. This allows us to use existing integer solvers. We give experiments that show how these preconditioning techniques can improve propagation performance on dense systems. 1

A class of ABS algorithms for Diophantine linear systems

Numerische Mathematik, 2001

Systems of integer linear (Diophantine) equations arise from various applications. In this paper we present an approach, based upon the ABS methods, to solve a general system of linear Diophantine equations. This approach determines if the system has a solution, generalizing the classical fundamental theorem of the single linear Diophantine equation. If so, a solution is found along with an integer Abaffian (rank deficient) matrix such that the integer combinations of its rows span the integer null space of the cofficient matrix, implying that every integer solution is obtained by the sum of a single solution and an integer combination of the rows of the Abaffian. We show by a counterexample that, in general, it is not true that any set of linearly independent rows of the Abaffian forms an integer basis for the null space, contrary to a statement by Egervary. Finally we show how to compute the Hermite normal form for an integer matrix in the ABS framework.

On generating all minimal integer solutions for a monotone system of linear inequalities

Automata, Languages …, 2001

We consider the problem of enumerating all minimal integer solutions of a monotone system of linear inequalities. We first show that for any monotone system of r linear inequalities in n variables, the number of maximal infeasible integer vectors is at most rn times the number of minimal integer solutions to the system. This bound is accurate up to a polylog(r) factor and leads to a polynomial-time reduction of the enumeration problem to a natural generalization of the well-known dualization problem for hypergraphs, in which dual pairs of hypergraphs are replaced by dual collections of integer vectors in a box. We provide a quasi-polynomial algorithm for the latter dualization problem. These results imply, in particular, that the problem of incrementally generating minimal integer solutions of a monotone system of linear inequalities can be done in quasi-polynomial time.

Trichotomy for integer linear systems based on their sign patterns

Discrete Applied Mathematics, 2016

In this paper, we consider solving the integer linear systems, i.e., given a matrix A ∈ R m×n , a vector b ∈ R m , and a positive integer d, to compute an integer vector x ∈ D n such that Ax ≥ b, where m and n denote positive integers, R denotes the set of reals, and D = {0, 1,. .. , d − 1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index η which is based only on the sign pattern of A. For a real γ, let ILS = (γ) denote the family of the problem instances I with η(I) = γ. We then show the following trichotomy: ILS = (γ) is linearly solvable, if γ < 1, ILS = (γ) is weakly NP-hard and pseudo-polynomially solvable, if γ = 1, and ILS = (γ) is strongly NP-hard, if γ > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.

Certificates of linear mixed integer infeasibility

Operations Research Letters, 2008

A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx ≤ d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point. The result also extends to the mixed integer setting.

An Analysis of Mixed Integer Linear Sets Based on Lattice Point Free Convex Sets

Mathematics of Operations Research, 2010

Split cuts are cutting planes for mixed integer programs whose validity is derived from maximal lattice point free polyhedra of the form S := {x : ! 0 ! ! Tx ! ! 0 +1 } called split sets. The set obtained by adding all split cuts is called the split closure, and the split closure is known to be