Super-linear Lower Bounds for CSP Non-Redundancy via Shrinking Instances

TL;DR

This paper introduces a new framework called hypergraph projections to analyze CSP non-redundancy, enabling super-linear lower bounds and automating the discovery of predicate relationships using SAT solvers.

Contribution

It recontextualizes gadget reductions in CSPs through hypergraph projections, allowing precise lower bounds and automated analysis with SAT solvers.

Findings

Identifies CSP predicates with near-critical non-redundancy levels.

Provides super-linear lower bounds for certain CSP predicates.

Automates gadget reduction discovery using SAT solvers.

Abstract

The non-redundancy (NRD) of a constraint satisfaction problem (CSP) is a combinatorial quantity closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlstr\"om [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with these previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.