Distance-sparsity transference for vertices of corner polyhedra

TL;DR

This paper establishes a new bound linking proximity and sparsity for vertices of corner polyhedra, improving proximity estimates in integer programming and connecting solution support size with minimal nonzero entries.

Contribution

It introduces a transference bound that unifies proximity and sparsity concepts, providing exponential improvements and new insights for integer linear programming solutions.

Findings

Exponential improvement on proximity bounds in knapsack problems

Connection between minimal nonzero entries and support size in integer programs

New theoretical bounds 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 gives an exponential (in the size of support of a solution) 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.