Sum-Of-Squares To Approximate Knapsack
Pravesh K. Kothari, Sherry Sarkar
TL;DR
This work analyzes the tight integrality gap of the sum-of-squares SDP for Knapsack via a pseudo-distribution perspective. It proves that any degree-$2t$ SoS pseudo-distribution satisfying the knapsack constraint yields $\tilde{\mathbb{E}}\left(\sum_i v_i x_i\right) \le \left(1 + \frac{1}{t-1}\right) OPT$, giving an integrality gap bound of the same form for the degree-$2t$ SDP. By encoding constraints through moment matrices and leveraging low-degree reductions (Red) and moment-linearities, the paper connects SoS relaxations to classic LP arguments while enabling tighter approximations. The results highlight the potential of SoS methods to design approximation algorithms for knapsack and related combinatorial problems by operating in the pseudo-distribution/degree framework. Overall, the findings demonstrate that higher-degree SoS relaxations can yield provably tighter guarantees than standard LP relaxations for NP-hard packing problems.
Abstract
These notes give a self-contained exposition of Karlin, Mathieu and Nguyen's tight estimate of the integrality gap of the sum-of-squares semidefinite program for solving the knapsack problem. They are based on a sequence of three lectures in CMU course on Advanced Approximation Algorithms in Fall'21 that used the KMN result to introduce the Sum-of-Squares method for algorithm design. The treatment in these notes uses the pseudo-distribution view of solutions to the sum-of-squares SDPs and only rely on a few basic, reusable results about pseudo-distributions.