Sum-Of-Squares To Approximate Knapsack

2025-02-18

AI Summary

This work investigates the integrality gap of the Sum-of-Squares (SoS) semidefinite programming hierarchy for the NP-hard knapsack problem. Using a pseudodistribution-based perspective, the paper concisely reconstructs and strengthens the classical Karlin–Mathieu–Nguyen result. Methodologically, it integrates SoS hierarchies, pseudodistribution modeling, combinatorial analysis, and probabilistic moment methods. The main contribution is a tight characterization: the integrality gap of the SoS-$k$ relaxation for knapsack equals exactly $1 - 1/sqrt{2pi k}$, and this bound is asymptotically tight. This establishes the first precise correspondence between SoS level $k$ and approximation quality for knapsack. Beyond resolving a long-standing open question, the result provides a foundational example for understanding the theoretical limits of SoS in combinatorial optimization. Moreover, it yields an interpretable and reusable pseudodistribution-driven framework for designing and analyzing approximation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Estimate integrality gap for knapsack
• Introduce Sum-of-Squares method
• Use pseudo-distribution view in solutions

Innovation

• Sum-of-Squares method
• pseudo-distribution view
• integrality gap estimate