The Richness of CSP Non-redundancy

TL;DR

This work develops a comprehensive theory of CSP non-redundancy (NRD), revealing that NRD can realize every rational growth exponent Θ(n^r) for appropriate predicates and exposing rich structure beyond linear bounds. Central to the advance are Catalan polymorphisms and Mal'tsev embeddings, which illuminate when NRD is linear and distinguish Abelian from non-Abelian embeddings (e.g., the PAULI predicate). The authors formulate a robust algebraic framework for conditional NRD via partial promise polymorphisms and patterns, linking NRD to hypergraph Turán problems and the Erdős box problem. They also establish a complete classification for binary conditional NRD, show infinite families of ternary NRD exponents approaching quadratic limits, and develop a powerful pattern-power operation that unifies reductions in conditional NRD. Collectively, the results deepen the connection between CSP non-redundancy, kernelization, and extremal combinatorics, and lay groundwork for future algebraic and Turán-type approaches to NRD in broader CSP settings.

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.

Figures (5)

• Figure 1: The 3-universal/cube pattern $U_3$ and the resulting partial Boolean function (named $u_3$). Each Boolean tuple is colored with the color of the identity from which it can be derived. The two constant Boolean tuples can be produced by any identity and are colored in black.
• Figure 2: The polymorphism pattern $U_3^2$ corresponding to the square of $U_3$ from Figure \ref{['fig:3_cube']}. For example, the first equation is obtained by combining the orange and blue equation in $U_3$. The corresponding 3-partite hypergraph is obtained by letting each partite set correspond to variables $a,b,c,d$ in the corresponding equation, and adding a hyperedge for each "column" of the equations.
• Figure 3: This figure illustrates how a bipartite graph $G$ with girth $6$ (left) is a non-redundant instance of $\operatorname{CSP}(C^*_6 \mid C_6)$ (right). In particular, the colors illustrate a homomorphism from $G$ to $C_6$ such that only the target edge $(x_0, y_0)$ is mapped to $(0,0)$. This coloring is generalized to bipartite graphs of arbitrary girth in \ref{['lem:cycle-nrd']}.
• Figure 4: A hypergraph representation of $\operatorname{CYC}_5$, with the red hyperedge being the one to delete to get $\operatorname{CYC}^*_5$. Note that if we delete any row from this figure, we get a graph isomorphic to $C_{10}$.
• Figure 5: Black edges are known results. Red edges are new results proved in this paper. Blue edges are conjectures. Dashed edges apply to Boolean predicates only. Note that the notion of an "Abelian embedding" is the same for finite and infinite groups due to a result of khanna2024Characterizations.

Theorems & Definitions (206)

• Theorem 1.1: Every fractional exponent as NRD
• Theorem 1.2: Infinitely many ternary NRD exponents approaching $2$
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1: Non-redundancy of a predicate bessiere2020Chain
• Definition 2.2: Non-redundancy of multipartite instances
• Lemma 2.3: Implicit in brakensiek2024Redundancy