Integrality Gaps of Linear and Semi-definite Programming Relaxations for Knapsack

TL;DR

This paper analyzes the integrality gaps of linear and semi-definite programming relaxations for the Knapsack problem, showing persistent gaps in Sherali-Adams and rapid gap closure in Lasserre hierarchy.

Contribution

It demonstrates that the Sherali-Adams hierarchy maintains a large integrality gap up to many rounds, while the Lasserre hierarchy effectively reduces the gap after a few rounds, using a novel decomposition theorem.

Findings

Sherali-Adams hierarchy has a gap of 2 - epsilon up to linear rounds.

Lasserre hierarchy reduces the gap to t/(t-1) after t rounds.

First use of more than a few rounds in Lasserre hierarchy for Knapsack.

Abstract

In this paper, we study the integrality gap of the Knapsack linear program in the Sherali- Adams and Lasserre hierarchies. First, we show that an integrality gap of 2 - {\epsilon} persists up to a linear number of rounds of Sherali-Adams, despite the fact that Knapsack admits a fully polynomial time approximation scheme [27,33]. Second, we show that the Lasserre hierarchy closes the gap quickly. Specifically, after t rounds of Lasserre, the integrality gap decreases to t/(t - 1). To the best of our knowledge, this is the first positive result that uses more than a small number of rounds in the Lasserre hierarchy. Our proof uses a decomposition theorem for the Lasserre hierarchy, which may be of independent interest.