Lower Bounds for CSP Hierarchies Through Ideal Reduction

TL;DR

This work develops a general method to derive level lower bounds for (promise) CSP hierarchies from degree lower bounds in algebraic proof systems by constructing integral pseudo-reduction operators. It shows that the Alekhnovich–Razborov framework can be used to fool the cohomological $k$-consistency algorithm, yielding asymptotically optimal lower bounds for $c$ vs. $\ell$-coloring and providing a streamlined proof for lax and null-constraining CSPs. The results illuminate a concrete bridge between proof complexity and CSP hierarchies, enabling transfer of degree-lower-bound techniques to hierarchy-level separations and highlighting the potential for applying these methods to additional hierarchies. Overall, the paper advances understanding of uniform CSP algorithms and presents a versatile toolkit for proving hierarchy lower bounds with implications for PCSPs and related combinatorial problems.

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].

Theorems & Definitions (64)

• Theorem 1.1: Informal
• Theorem 1.2: Informal, cf. CN25Hierarchies, Theorem 1.3
• Lemma 2.1: Boolean Nullstellensatz
• Definition 2.2: Pseudo-reduction operator
• Lemma 2.3: Razborov98LowerBound, Lemma 3.2
• proof : Proof sketch
• Lemma 3.1
• proof
• Remark 3.2
• Definition 4.1: Size-$D$ closure operator