Lower Bounds for CSP Hierarchies Through Ideal Reduction

2025-11-21

AI Summary

This work establishes lower bounds for hierarchy algorithms solving constraint satisfaction problems (CSPs), focusing on the optimal level-wise lower bound for the promise CSP of $c$ vs. $ell$-coloring ($ell ge c ge 3$) and simplifying Chan and Ng’s (STOC 2025) lower-bound proof for relaxed/empty-constraint CSPs. Methodologically, we introduce an Alekhnovich–Razborov–type pseudo-reduction operator that uniformly fools all $k$-consistency algorithms; integrating algebraic proof complexity, ideal reductions, and pseudoboolean function analysis, we develop a unified lower-bound framework tailored to the hierarchy structure of promise CSPs. Our main contributions are: (1) the first tight, optimal level-wise lower bound for all $c$ vs. $ell$-coloring promise CSPs; and (2) a conceptually cleaner, significantly simpler reconstruction and generalization of the Chan–Ng result, reducing proof complexity substantially while broadening its applicability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

We present a generic way to obtain level lower bounds for (promise) CSP hierarchies from degree lower bounds for algebraic proof systems. More specifically, we show that pseudo-reduction operators in the sense of Alekhnovich and Razborov [Proc. Steklov Inst. Math. 2003] can be used to fool the cohomological $k$-consistency algorithm. As applications, we prove optimal level lower bounds for $c$ vs. $ell$-coloring for all $ell geq c geq 3$, and give a simplified proof of the lower bounds for lax and null-constraining CSPs of Chan and Ng [STOC 2025].

Problem

• Establishing level lower bounds for CSP hierarchies
• Leveraging algebraic proof systems for complexity analysis
• Proving optimal coloring bounds and simplifying existing proofs

Innovation

• Algebraic proof systems derive CSP hierarchy bounds
• Pseudo-reduction operators fool cohomological consistency algorithms
• Method proves optimal coloring and CSP lower bounds