Sum-Of-Squares To Approximate Knapsack

TL;DR

This paper explains how the sum-of-squares semidefinite programming approach can tightly approximate the knapsack problem, providing insights into its integrality gap and solution structure.

Contribution

It offers a self-contained exposition of the Karlin, Mathieu, and Nguyen result on the sum-of-squares integrality gap for knapsack, using pseudo-distributions and accessible techniques.

Findings

Tight estimate of the sum-of-squares integrality gap for knapsack

Use of pseudo-distributions simplifies understanding of SOS solutions

Provides educational material for advanced approximation algorithms

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.