Integrality Gaps of Integer Knapsack Problems

TL;DR

This paper investigates the integrality gaps in integer knapsack problems, providing bounds and demonstrating that typical cases have significantly smaller gaps than worst-case scenarios.

Contribution

It establishes optimal bounds for integrality gaps and shows that average-case gaps are much smaller than worst-case bounds in integer knapsack problems.

Findings

Optimal bounds for integrality gaps are derived.

Typical problem instances have smaller gaps than worst-case scenarios.

Randomized analysis reveals significant difference between average and worst-case gaps.

Abstract

We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario.