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 P (a, b) = {x ∈ R n ≥0 : a ⊤ x = b}, a ∈ Z n >0 , there exists an integer point z * ∈ P (a, b) such that, denoting by s the size of the support of z * and assuming s > 0,where • ∞ stands for the ℓ∞-norm. The bound 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.

Finding an Optimal Proximity Bound in a Very Special Scenario

“…Recall that Λ denotes the affine lattice in R m+1 containing all integer points which satisfy Ax = b. In particular, its projection π(Λ) is a one-dimensional affine lattice which, in light of [1,Lemma 10], has determinant…”

“…We will estimate the ℓ ∞ -distance from a vertex x * of the polyhedron P , which is given by a basis γ as in (1), to the set of its (feasible) lattice points z ∈ P ∩ Z n . It is worth noting that bounds of this form provide information regarding the level of accuracy when one "relaxes" an integer program (IP) and instead solves the related linear program (LP).…”

Improving the Cook et al. Proximity Bound Given Integral Valued Constraints

“…. , J r be a sequence of feasible bases of this system, with corresponding basic feasible solutions 0 = y (0) , y (1) , . .…”

“…Further, recall that S (x * ) ⊆ P. Let N := λ 1 + • • • + λ t . We choose a sequence 0 = x (0) , x (1) , . .…”

“…This leads to the proximity question that we consider: For a given matrix A, find a value π ≥ 0 such that . Actually, under this assumption there are many results indicating that proximity is independent of n: Aliev et al [2] prove that proximity is upper bounded by the largest entry of A for knapsack polytopes, Veselov and Chirkov's result [19] implies a proximity bound of 2 when ∆ n (A) ≤ 2, and Aliev et al [1] prove a bound of ∆ n (A) for corner polyhedra.…”