Additive Sparsification of CSPs

TL;DR

This paper proves that all Boolean and finite-domain CSPs can be approximated by additive sparsifiers, simplifying the structure needed for efficient approximation algorithms without edge weights.

Contribution

It introduces the concept of additive sparsification for CSPs and proves that such sparsifiers exist for all Boolean and finite-domain predicates.

Findings

All Boolean CSPs admit additive sparsifiers.

Finite-domain CSPs also admit additive sparsifiers under a new all-but-one sparsification notion.

The results extend sparsification techniques to a broader class of problems.

Abstract

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 $D$.