Multiplicative cut sparsifiers, introduced by Benczúr 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 Živný [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→ { 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→ { 0,1} of a fixed arity k on an arbitrary finite domain D.

Benczúr 和 Karger [STOC'96] 引入的乘法切割稀疏器已被证明具有极大的影响力并找到了各种应用。Filtser 和 Krauthgamer [SIDMA'17] 在布尔域上使用其他 2 变量谓词建立了图的稀疏性，Butti 和 Živný [SIDMA'20] 在非布尔域上建立了精确的特征描述。

Bansal、Svensson 和 Trevisan [FOCS'19] 引入了一种较弱的稀疏化概念，称为“加法稀疏化”，它不需要图边缘的权重。特别是班萨尔等人。设计了用于图和超图切割的加性稀疏器的算法。

作为我们的主要结果，我们确定所有布尔约束满足问题（CSP）都承认可加稀疏器；也就是说，对于固定数量k的每个布尔谓词P :{ 0,1} ^k → { 0,1} ，我们证明 CSP( P ) 允许加法稀疏器。根据我们新引入的非布尔谓词的除一稀疏化概念，我们证明 CSP( P ) 允许任何谓词P的加法稀疏器：D ^k → { 0,1} 具有任意的固定数量k有限域D .