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

Abstract

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 − 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). This answers the open question in [10]. Also, 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.

Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP

“…called the level, where the level 0 tightening means the initial LP. A key "decomposition theorem" (see Theorem 4.1, [13,19]) asserts that a feasible solution at level t can be written as a convex combination of feasible solutions at a lower level such that all of these lower-level solutions y are "locally integral." Here, "locally integral" means that there is a specified subset J ⊆ E(G) − E(T ), such that the solution y takes only zero or one values on this subset (i.e., y e ∈ {0, 1}, ∀e ∈ J ).…”

“…In fact, our results can be stated and proved without going into the formalities of the Lasserre system. Essentially, we apply one well-known result about the Lasserre system, namely, the decomposition theorem of Karlin-Mathieu-Nguyen, see [13,19,20]. The comprehensive recent survey by Rothvoß [20] presents this result and much more.…”

“…Now, consider our LP (L P 0 ) for TAP, and let Las t (L P 0 ) denote the level t tightening of (L P 0 ) by the Lasserre system. 3 Rothvoß, see [19,Theorem 2], formulated the following decomposition theorem for the Lasserre system, based on an earlier decomposition theorem due to Karlin-Mathieu-Nguyen [13]. (We use this particular formulation and not the original statement of [13]; hence, we reference both [13] and [19].)…”

“…3 Rothvoß, see [19,Theorem 2], formulated the following decomposition theorem for the Lasserre system, based on an earlier decomposition theorem due to Karlin-Mathieu-Nguyen [13]. (We use this particular formulation and not the original statement of [13]; hence, we reference both [13] and [19].) Theorem 4.1 Let J ⊆ E. Let k be a positive integer such that |ones(x) ∩ J | ≤ k for every feasible solution x of (L P 0 ).…”

Untitled

“…Sherali-Adams relaxation is one of the lift-and-project schemes which have received much interest recently in the approximation algorithms community (see, e.g [21] and references therein). A question of particular interest is how the integrality gaps evolve through a series of rounds of Sherali-Adams lift-and-project operations.…”