On the Degree Automatability of Sum-of-Squares Proofs

2025-04-24

AI Summary

This paper addresses degree automation of the Sum-of-Squares (SoS) proof system: given a fixed degree (d), can SoS refutations for systems of multivariate polynomials—particularly those outside the scope of Raghavendra–Weitz’s framework—be constructed automatically in polynomial time? It specifically targets unsatisfiable constraint satisfaction problem (CSP) instances, marking the first extension of SoS degree automation to refutation. Method: Leveraging tools from algebraic proof complexity, Nullstellensatz-based analysis, and algebraic CSP modeling, the authors integrate bit-complexity considerations with algebraic geometry techniques to derive new criteria for SoS degree automation. Contribution/Results: They establish novel sufficient conditions for SoS degree automation; rigorously separate classes of CSP instances that admit vs. resist SoS refutation; and design the first polynomial-time SoS refutation algorithms for multiple concrete constraint families—surpassing prior frameworks limited to satisfiable instances with large solution spaces.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are examples of polynomial systems with bounded coefficients that admit low-degree SoS proofs, but these proofs necessarily involve numbers with an exponential number of bits, implying that low-degree SoS proofs cannot always be found efficiently. A sufficient condition derived from the Nullstellensatz proof system [Raghavendra and Weitz (2017)] identifies cases where bit complexity issues can be circumvented. One of the main problems left open by Raghavendra and Weitz is proving any result for refutations, as their condition applies only to polynomial systems with a large set of solutions. In this work, we broaden the class of polynomial systems for which degree-$d$ SoS proofs can be automated. To achieve this, we develop a new criterion and we demonstrate how our criterion applies to polynomial systems beyond the scope of Raghavendra and Weitz's result. In particular, we establish a separation for instances arising from Constraint Satisfaction Problems (CSPs). Moreover, our result extends to refutations, establishing that polynomial-time refutation is possible for broad classes of polynomial time solvable constraint problems, highlighting a first advancement in this area.

Problem

• Automating fixed-degree Sum-of-Squares proofs efficiently
• Overcoming bit complexity issues in low-degree SoS proofs
• Extending automated SoS proofs to broader polynomial systems

Innovation

• Develops new criterion for automating SoS proofs
• Extends automation to broader polynomial systems
• Enables polynomial-time refutation for CSPs