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

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.

💡 Research Summary

The paper investigates how two powerful hierarchies of relaxations—Sherali‑Adams (SA) and Lasserre (LAS)—affect the integrality gap of the classic 0‑1 knapsack problem. Although knapsack admits a fully polynomial‑time approximation scheme (FPTAS), its natural linear programming (LP) relaxation can be far from optimal, prompting the study of systematic strengthening procedures.

Sherali‑Adams results.
The authors construct a family of knapsack instances in which, even after Θ(n) rounds of the SA hierarchy, the integrality gap remains at least 2 − ε for any fixed ε > 0. The construction builds on “pseudo‑instance” techniques: item weights and profits are carefully scaled so that each additional SA round only introduces a limited set of higher‑order linear constraints, insufficient to eliminate the fractional solution that achieves roughly twice the optimal integer profit. The proof shows that the SA hierarchy, despite adding polynomially many linear constraints per round, cannot capture the combinatorial structure that forces items to be either fully taken or fully discarded. Consequently, SA fails to close the gap even when the number of rounds grows linearly with the problem size.

Lasserre results.
In stark contrast, the Lasserre hierarchy quickly shrinks the gap. The authors prove that after t rounds of LAS, the integrality gap is bounded by t / (t − 1). The key technical contribution is a new decomposition theorem for the Lasserre hierarchy. This theorem states that any feasible moment matrix after t rounds can be expressed as a convex combination of moment matrices that are supported on a small “core” set of at most t variables. By fixing the remaining variables to either 0 or 1 (or averaging over them), the original large SDP reduces to a collection of much smaller SDPs, each of which already satisfies a strong integrality property. The analysis shows that each reduced sub‑problem yields a solution whose value is at most (t − 1)/t of the optimal fractional value, leading directly to the overall gap bound. As t increases, the bound converges to 1, meaning that a modest number of Lasserre rounds essentially eliminates the integrality gap.

Implications and novelty.
The paper’s two main contributions are (1) a negative result that highlights the limitations of linear‑only hierarchies for knapsack, and (2) a positive result demonstrating that semidefinite‑based hierarchies can close the gap with surprisingly few rounds. The decomposition theorem is of independent interest; it provides a systematic way to break down high‑level Lasserre solutions into tractable components, a technique that may be applicable to other combinatorial optimization problems. Moreover, this work is, to the best of the authors’ knowledge, the first to show a concrete, non‑trivial improvement in integrality gap after more than a constant number of Lasserre rounds for a classic NP‑hard problem.

Structure of the paper.
Section 1 reviews knapsack, its LP relaxation, and the definitions of SA and LAS hierarchies. Section 2 presents the SA lower‑bound construction, detailing the instance parameters and the inductive argument that the gap persists through linear rounds. Section 3 introduces the Lasserre decomposition theorem, proves it, and derives the t/(t‑1) gap bound. Section 4 compares the two hierarchies, discusses why semidefinite constraints are fundamentally stronger for knapsack, and speculates on algorithmic consequences. Section 5 outlines future directions, including extending the decomposition approach to other packing and covering problems and exploring trade‑offs between the number of rounds and computational effort.

In summary, while the Sherali‑Adams hierarchy fails to substantially improve the knapsack integrality gap even after many rounds, the Lasserre hierarchy achieves rapid convergence, reducing the gap to near‑optimal after only a few rounds. This dichotomy underscores the power of semidefinite programming in hierarchy‑based relaxations and opens new avenues for designing tighter approximations for knapsack and related combinatorial optimization problems.