Distances to Lattice Points in Knapsack Polyhedra

TL;DR

This paper establishes optimal bounds on the distance from vertices of knapsack polyhedra to the nearest lattice points, showing average improvements and implications for the integrality gap in integer programming.

Contribution

It provides the first optimal upper bounds for lattice distances in knapsack polyhedra and analyzes the typical case, improving understanding of the integrality gap.

Findings

Optimal upper bounds for maximum-norm distances

Average-case improvements in lattice distances

Bounds on the integrality gap in typical knapsack problems

Abstract

We give an optimal upper bound for the maximum-norm 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.