The seminal work of Bencz\'ur and Karger demonstrated cut sparsifiers of near-linear size, with several applications throughout theoretical computer science. Subsequent extensions have yielded sparsifiers for hypergraph cuts and more recently linear codes over Abelian groups. A decade ago, Kogan and Krauthgamer asked about the sparsifiability of arbitrary constraint satisfaction problems (CSPs). For this question, a trivial lower bound is the size of a non-redundant CSP instance, which admits, for each constraint, an assignment satisfying only that constraint (so that no constraint can be dropped by the sparsifier). For graph cuts, spanning trees are non-redundant instances. Our main result is that redundant clauses are sufficient for sparsification: for any CSP predicate R, every unweighted instance of CSP(R) has a sparsifier of size at most its non-redundancy (up to polylog factors). For weighted instances, we similarly pin down the sparsifiability to the so-called chain length of the predicate. These results precisely determine the extent to which any CSP can be sparsified. A key technical ingredient in our work is a novel application of the entropy method from Gilmer's recent breakthrough on the union-closed sets conjecture. As an immediate consequence of our main theorem, a number of results in the non-redundancy literature immediately extend to CSP sparsification. We also contribute new techniques for understanding the non-redundancy of CSP predicates. In particular, we give an explicit family of predicates whose non-redundancy roughly corresponds to the structure of matching vector families in coding theory. By adapting methods from the matching vector codes literature, we are able to construct an explicit predicate whose non-redundancy lies between $\Omega(n^{1.5})$ and $\widetilde{O}(n^{1.6})$, the first example with a provably non-integral exponent.

翻译：Benczúr与Karger的开创性工作证明了近线性规模的割稀疏化器在理论计算机科学领域具有多重应用价值。后续研究将稀疏化推广至超图割以及近期阿贝尔群上的线性编码。十年前，Kogan与Krauthgamer提出了关于任意约束满足问题（CSP）可稀疏化性的疑问。针对该问题，一个平凡下界是非冗余CSP实例的规模，此类实例允许每个约束存在仅满足该约束的赋值（从而稀疏化器无法删除任何约束）。在图割问题中，生成树即为非冗余实例。我们的核心结论是冗余子句足以实现稀疏化：对于任意CSP谓词R，每个CSP(R)的无权实例均存在规模至多为其非冗余度（相差多对数因子）的稀疏化器。对于有权实例，我们同样将可稀疏化性精确归结为谓词的链长特性。这些结果精确界定了任意CSP可稀疏化的程度。本研究的关键技术要素是熵方法在吉尔默近期攻克并封闭集猜想突破成果中的创新应用。作为主定理的直接推论，非冗余性研究领域的多项成果可立即推广至CSP稀疏化。我们还提出了理解CSP谓词非冗余性的新技术，特别构建了一个谓词显式族，其非冗余度大致对应于编码理论中匹配向量族的结构。通过适配匹配向量编码文献的方法，我们成功构造出非冗余度介于$\Omega(n^{1.5})$与$\widetilde{O}(n^{1.6})$之间的显式谓词，这是首个被证明具有非整数指数的实例。