The Reach of Abelian Covers in Hypergraphs

2026-09-29

AI Summary

This study addresses the previously unclear dependency-capturing capabilities of Abelian covers of hypergraphs and the characterization of non-redundancy in constraint satisfaction problems (CSPs). To this end, it formally defines and distinguishes two novel classes of covers—Abelian and Catalan—for the first time. By integrating lattice theory, algebraic topology, and nilpotent group theory, the work precisely delineates the capability limits and boundaries of these covers in deriving dependencies. Furthermore, it establishes that ternary CSPs exhibit linear non-redundancy, thereby providing a near-optimal theoretical foundation for streaming algorithms, sparsification, and kernelization.

Abstract

Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers--which check if each vertex appears in an even number of hyperedges--have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on the non-redundancy of constraint satisfaction problems (CSPs), we introduce and study two novel families of covers of hypergraphs which are stricter than even covers: \emph{Abelian} covers and Catalan covers. Abelian covers are similar to even covers, except that arithmetic is now done over the integers rather than modulo 2, allowing us to capture dependences over arbitrary Abelian groups. Catalan covers capture the behavior of non-Abelian groups by only allowing local cancellations in a sequence of hyperedges. We prove three main results about Abelian and Catalan covers. First, using tools from lattice theory, we show that any $r$-uniform hypergraph with $n$ vertices and $n \log(r)$ hyperedges has an Abelian cover. Second, using tools from algebraic topology, we show that in any $3$-uniform hypergraph, Abelian covers and Catalan covers are equivalent; thereby showing that Catalan covers emerge after $O(n)$ hyperedges in $3$-uniform hypergraphs. Finally, using the theory of nilpotent groups, we show that there exists a $4$-uniform hypergraph which has an Abelian cover but not a Catalan cover. Collectively, these results exactly characterize the reach that Abelian covers have in deducing dependences in hypergraphs. As our primary application, we show that any arity-$3$ CSP with an infinite-domain Mal'tsev extension has linear non-redundancy. This implies near optimal streaming, sparsification, and kernelization algorithms for this family of CSPs. Previously, such a result was only known for the much simpler case of arity-$2$ CSPs.

Problem

• Hypergraph covers
• Abelian covers
• Catalan covers
• Constraint satisfaction problems
• Non-redundancy

Innovation

• Hypergraph Covers
• Abelian Covers
• Catalan Covers
• Constraint Satisfaction Problems
• Algebraic Topology