Non-redundancy, introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020), is a structural parameter for Constraint Satisfaction Problems ($\mathsf{CSPs}$) that governs kernelization, exact and approximate sparsification, and exact streaming complexity. It is the largest size of a $\mathsf{CSP}$ instance admitting no smaller subinstance with the same satisfying assignments. We study non-redundancy $\mathsf{NRD}_n(R)$ for Boolean symmetric $\mathsf{CSPs}$ defined by an $r$-ary relation $R$ whose value depends only on Hamming weight. An instance of $\mathsf{CSP}(R)$ has $n$ variables and constraints given by $r$-tuples; a constraint is satisfied exactly when the induced tuple lies in $R$. This class includes natural predicates such as cuts and $k$-SAT clauses. Our main result is a near-complete classification of the asymptotic growth of $\mathsf{NRD}_n(R)$ for symmetric Boolean predicates of arity at most $5$. Using computational experiments and algebraic upper- and lower-bound criteria, we resolve every predicate of arity at most $4$ and all but two predicates of arity $5$. For upper bounds, we introduce $t$-balancedness, a lifted, higher-degree version of the balancedness notion of Chen, Jansen, and Pieterse (Algorithmica 2020). We prove that $t$-balancedness is equivalent to the existence of degree-$t$ multilinear polynomials capturing $R$, and hence implies $\mathsf{NRD}_n(R)=O(n^t)$. For lower bounds, we use Carbonnel's (CP 2022) framework: predicates admitting a special reduction from $k$-ary OR inherit OR's lower bound $Ω(n^k)$. The only unresolved arity-$5$ predicates in our framework have bounds $Ω(n^2)$ and $O(n^3)$; we reduce their exact classification to natural extremal set-system questions.

翻译：非冗余性（non-redundancy）由Bessiere、Carbonnel和Katsirelos（AAAI 2020）提出，是约束满足问题（$\mathsf{CSPs}$）的一种结构参数，用于刻画核化、精确与近似稀疏化以及精确流计算复杂度。其定义为：不存在具有相同可满足赋值且规模更小的子实例的$\mathsf{CSP}$实例的最大规模。本文研究由$r$元关系$R$定义的布尔对称$\mathsf{CSP}$的非冗余性$\mathsf{NRD}_n(R)$，其中$R$的取值仅依赖于汉明重量。$\mathsf{CSP}(R)$实例包含$n$个变量，其约束由$r$元组给出；当且仅当所涉及元组属于$R$时，约束被满足。该类别涵盖自然谓词（如割和$k$-SAT子句）。我们的主要结果是对至多5元对称布尔谓词的$\mathsf{NRD}_n(R)$渐近增长性给出近乎完整的分类。通过计算实验及代数上下界准则，我们解决了所有至多4元谓词以及5元谓词中除两个外的所有情形。对于上界，我们引入$t$-平衡性（$t$-balancedness），即Chen、Jansen与Pieterse（Algorithmica 2020）平衡性概念的高次提升版本。我们证明$t$-平衡性等价于存在捕获$R$的$t$次多元多项式，从而蕴含$\mathsf{NRD}_n(R)=O(n^t)$。对于下界，我们采用Carbonnel（CP 2022）框架：允许从$k$元OR进行特殊规约的谓词继承OR的下界$\Omega(n^k)$。我们框架中唯一未解决的5元谓词其界分别为$\Omega(n^2)$和$O(n^3)$；我们将这些谓词的精确分类归约为自然极值集系统问题。