Improving proximity bounds using sparsity

Abstract

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

Distance-sparsity transference for vertices of corner polyhedra

“…Applying Theorem 1 with the vertex z * ∈ P (a, b) we immediately obtain ( 14) and (15). Further, the bound (6) implies for r ≥ 2 the non-strict inequality (45)…”

“…More recent contributions include results of Eisenbrand and Shmonin [12] and Aliev et al [3,2,1]. Further, in a very recent work Lee, Paat, Stallknecht and Xu [15] apply new sparsity-type bounds to refine the bounds for proximity.…”