Integrality Gaps of Linear and Semi-Definite Programming Relaxations for Knapsack
نویسندگان
چکیده
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.
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