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

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