On Approximability of Satisfiable k-CSPs: V

2024-08-27

Electron. Colloquium Comput. Complex.

AI Summary

This work addresses the approximability of Max-CSP, establishing for the first time an algorithmic hardness tightness framework applicable to **all satisfiable k-CSP instances**, thereby overcoming Raghavendra’s restriction to nearly satisfiable instances. We introduce the **mixed invariance principle**, which systematically links third-order correlations in discrete domains to expectations over hybrid Gaussian/Abelian group spaces—a novel connection. Our method combines Gaussian elimination with semidefinite programming to yield a hybrid approximation algorithm and constructs a perfectly complete “dictator vs. pseudorandom” test. For a broad class of predicates, we achieve optimal approximation ratios: the algorithm’s performance exactly matches the Unique Games Conjecture (UGC)-based hardness lower bounds, yielding tightness. This resolves the approximability threshold for these CSPs under UGC, unifying algorithm design and hardness analysis across the full spectrum of satisfiable instances.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.

Problem

• Extending algorithm-hardness framework for all Max-CSPs
• Developing hybrid algorithm combining Gaussian elimination and SDP
• Introducing mixed invariance principle for correlation analysis

Innovation

• Hybrid algorithm combines Gaussian elimination and SDP relaxation
• Introduces mixed invariance principle for correlation analysis
• Extends Raghavendra's framework with perfect completeness tests