Additive Sparsification of CSPs

Abstract

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.

Additive Sparsification of CSPs

“…As mentioned in Section 1, we begin by observing that Theorem 3 also works for directed hypergraphs. The simple proof of this fact can be found in the full version [29]. (We emphasise that we treat k as a constant, cf.…”

“…We refer the reader to a nice recent survey of Elkin and Neimain for more details [19]. Some of the proofs are deferred to the full version of this paper [29].…”

“…To complete the picture for even k, we reduce to Proposition 13 by a simple "complementarity trick"; the proof can be found in the full version [29].…”

“…The final piece in the jigsaw, proved in in the full version [29], shows how to reduce sparsification of k-uniform hypergraphs with k odd to the case of (k + 1)-uniform hypergraphs by adding a universal vertex and extending the original predicate by one dimension. ▶ Proposition 16.…”

“…Consider the predicate Cut : [q] 2 → {0, 1} defined by Cut(x, y) = 1 ⇐⇒ x ̸ = y. A simple (but lengthy) argument in the full version [29] shows that cliques do not have a Cut-sparsifier using such a definition. In fact, the same argument works for any predicate P with P (q − 2, q − 1) = P (q − 1, q − 2) = 1 and P (q − 2, q − 2) = P (q − 1, q − 1) = 0.…”