Multiplicative cut sparsifiers, introduced by Bencz\'ur and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and \v{Z}ivn\'y [SIDMA'20]. Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate $P:\{0,1\}^k\to\{0,1\}$ of a fixed arity $k$, we show that CSP($P$) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP($P$) admits an additive sparsifier for any predicate $P:D^k\to\{0,1\}$ of a fixed arity $k$ on an arbitrary finite domain $D$.

翻译：乘法割稀疏化由Benczúr和Karger [STOC'96]提出，已被证明极具影响力并广泛应用于多个领域。Filtser与Krauthgamer [SIDMA'17]刻画了布尔域上其他二变量谓词图的稀疏化精确特征，而Butti与Živný [SIDMA'20]将其推广至非布尔域。Bansal、Svensson和Trevisan [FOCS'19]引入了一种更弱的稀疏化概念，称为“加性稀疏化”，该概念不需要对图的边赋予权重。具体而言，Bansal等人设计了针对图和超图中割的加性稀疏化算法。作为我们的主要结果，我们证明所有布尔约束满足问题（CSP）均存在加性稀疏化：即对于固定元数$k$的任意布尔谓词$P:\{0,1\}^k\to\{0,1\}$，我们证明CSP($P$)存在加性稀疏化。基于我们针对非布尔谓词新提出的“除一以外全部抑制”稀疏化概念，我们证明对于任意有限域$D$上固定元数$k$的任意谓词$P:D^k\to\{0,1\}$，CSP($P$)均存在加性稀疏化。