In this paper, we study the integrality gap of the Knapsack linear program in the SheraliAdams and Lasserre hierarchies. First, we show that an integrality gap of 2 − ǫ 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.
Karlin, Mathieu, and Nguyen show that 1/ε rounds of Lasserre SDP gives a (1 + ε) approximation to the Knapsack problem REF .
Integrality Gaps of Linear and Semi-definite Programming Relaxations for Knapsack
ArXiv
ArXiv
Mathematics Computer Science