Distance-sparsity transference for vertices of corner polyhedra

Abstract

We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it implies that for any vertex x * of an integer feasible knapsack polytope, there exists an integer point z * ∈ P (a, b) such that, denoting by s the size of support of z * and assuming s > 0,(1)where • ∞ stands for the ℓ∞-norm. The bound (1) gives an exponential in s improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support.