On the Bit Size of Sum-of-Squares Proofs for Symmetric Formulations

2025-09-08

AI Summary

This work investigates whether symmetry inherently causes exponential growth in the bit-width of coefficients within Sum-of-Squares (SoS) proofs, thereby undermining computational feasibility. Focusing on Archimedean symmetric polynomial optimization problems, we integrate the SoS hierarchy, semidefinite programming relaxations, and group representation-theoretic symmetry analysis. We establish the first separation of two orthogonal sources of SoS complexity: proof degree and coefficient bit-size. Our results show that while symmetry may necessitate higher-degree SoS proofs, it does *not* intrinsically induce large-bit coefficients; low-degree symmetric SoS proofs admit compact numerical representations. This refutes the common assumption that symmetry inevitably leads to coefficient blow-up. The findings provide a new theoretical foundation for automated SoS solving and lower-bound analysis, advancing our understanding of computational tractability in symmetric algebraic proof systems. (149 words)

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

The Sum-of-Squares (SoS) hierarchy is a powerful framework for polynomial optimization and proof complexity, offering tight semidefinite relaxations that capture many classical algorithms. Despite its broad applicability, several works have revealed fundamental limitations to SoS automatability. (i) While low-degree SoS proofs are often desirable for tractability, recent works have revealed they may require coefficients of prohibitively large bit size, rendering them computationally infeasible. (ii) Prior works have shown that SoS proofs for seemingly easy problems require high-degree. In particular, this phenomenon also arises in highly symmetric problems. Instances of symmetric problems-particularly those with a small number of constraints-have repeatedly served as benchmarks for establishing high-degree lower bounds in the SoS hierarchy. It has remained unclear whether symmetry can also lead to large bit sizes in SoS proofs, potentially making low-degree proofs computationally infeasible even in symmetric settings. In this work, we resolve this question by proving that symmetry alone does not lead to large bit size SoS proofs. Focusing on symmetric Archimedean instances, we show that low-degree SoS proofs for such systems admit compact, low bit size representations. Together, these results provide a conceptual separation between two sources of SoS hardness-degree and bit size-by showing they do not necessarily align, even in highly symmetric instances. This insight guides future work on automatability and lower bounds: symmetry may necessitate high-degree proofs, but it does not by itself force large coefficients.

Problem

• Investigating bit size requirements in symmetric Sum-of-Squares proofs
• Resolving whether symmetry causes large coefficients in SoS proofs
• Separating degree and bit size as sources of SoS hardness

Innovation

• Symmetric Archimedean instances enable compact proofs
• Low-degree SoS proofs admit low bit size
• Symmetry separates degree and bit size hardness