Lower Bounds for CSP Hierarchies Through Ideal Reduction

TL;DR

This paper introduces a method to derive lower bounds for CSP hierarchies by leveraging algebraic proof system degree bounds, demonstrating optimal bounds for certain coloring problems and simplifying existing lower bound proofs.

Contribution

It establishes a generic reduction technique connecting algebraic proof degree bounds to CSP hierarchy lower bounds, and applies it to specific coloring and CSP problems.

Findings

Proves optimal level lower bounds for c vs. l-coloring for all l ≥ c ≥ 3.

Provides a simplified proof for lower bounds on lax and null-constraining CSPs.

Shows pseudo-reduction operators can fool the cohomological k-consistency algorithm.

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].