The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by A ∈ Z m×n and present an algorithm to solve such problems in polynomial-time provided that both the largest absolute value of an entry in A and m are constant. Then, this is applied to solve integer programs in inequality form in polynomial-time, where the absolute values of all maximal sub-determinants of A lie between 1 and a constant.

Abstract. We present structural results on solutions to the Diophantine system Ay = b, y ∈ Z t ≥0 that have the smallest number of nonzero entries. Our tools are algebraic and number theoretic in nature and include Siegel's lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

The support of a vector is the number of nonzero-components. We show that given an integral m × n matrix A, the integer linear optimization problem max c T x : Ax = b, x ≥ 0, x ∈ Z n has an optimal solution whose support is bounded by 2m log(2 √ m A ∞ ), where A ∞ is the largest absolute value of an entry of A. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.

We give an optimal upper bound for the ℓ∞-distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower bound.

Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the ℓ 0 -norm. Our main results are improved bounds on the ℓ 0norm of sparse solutions to systems Ax = b, where A ∈ Z m×n , b ∈ Z m and x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In the lattice case and certain scenarios of the semigroup case, we give polynomial time algorithms for computing solutions with ℓ 0 -norm satisfying the obtained bounds.