  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.
