The Richness of CSP Non-redundancy

2025-07-10

AI Summary

This paper investigates *non-redundancy degree (NRD)* of predicates in constraint satisfaction problems (CSPs)—the asymptotic growth rate, as a function of variable count $n$, of the maximum number of non-redundant clause instances a predicate can support. Addressing the central question “Which predicates exhibit $Theta(n^r)$ NRD?”, we establish: (i) for every rational $r geq 1$, there exists a finite predicate achieving exactly $Theta(n^r)$ NRD; (ii) a complete characterization of conditional NRD for all binary predicates; and (iii) a novel Mal’tsev embedding based on the quantum Pauli group, circumventing limitations of prior abelian-group embeddings. Methodologically, we integrate high-girth graph constructions, Carbonnel’s algebraic framework, and generalized Mal’tsev embeddings to develop a systematic algebraic theory of conditional non-redundancy. Our results uncover deep connections between NRD and sparsification, kernelization, and query complexity, and reveal a rich structural hierarchy of NRD within the polynomial hierarchy.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $Θ(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.

Problem

• Characterize non-redundancy in CSP predicates across variable counts
• Classify conditional non-redundancy for binary predicates via extremal graph theory
• Develop algebraic theory for conditional non-redundancy and Mal'tsev embeddings

Innovation

• Classifies binary predicates via high-girth graphs
• Develops algebraic theory for conditional non-redundancy
• Introduces Mal'tsev embeddings beyond Abelian groups