Sparsification of Binary CSPs

Butti, Silvia; Živný, Stanislav

doi:10.1137/19m1242446

Cited by 3 publications

(10 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The goal of our work is to understand how the notion of additive sparsification developed in [7] for cuts behaves on (hyper)graphs with other predicates (beyond cuts), deriving inspiration from the generalisations of cuts to other predicates in the multiplicative setting established in [20,14]. In particular, already Boolean binary predicates include interesting predicates such as the uncut edges (using the predicate P (x, y) = 1 iff x = y), covered edges (using the predicate P (x, y) = 1 iff x = 1 or y = 1), or directed cut edges (using the predicate P (x, y) = 1 iff x = 0 and y = 1).…”

Section: Motivationmentioning

confidence: 99%

Additive Sparsification of CSPs

Pelleg,

Živný

2023

ACM Trans. Algorithms

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

Section: Motivationmentioning

confidence: 99%

Additive Sparsification of CSPs

Pelleg,

Živný

2023

ACM Trans. Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, we use linear algebra to prove the correctness of the reduction. While the reduction via the k-partite k-fold cover was used in previous works on multiplicative sparsification [20,14], the subsequent non-trivial linear-algebraic analysis (Proposition 14) is novel and constitutes our main technical contribution, as well as our result that, unlike in the multiplicative setting, all Boolean predicates can be (additively) sparsified. We also show that our results immediately apply to the more general setting where different hyperedges are associated with different predicates (cf.…”

Section: Contributionsmentioning

confidence: 99%

“…Comparison to previous work As mentioned above, our sparsifiability result is obtained by a reduction via the k-partite k-fold cover to the cut case established in [7]. A reduction via the k-partite k-fold cover was also used (for k = 2) in previous work on multiplicative sparsification [20,14]. In particular, the correctness of the reduction for Boolean binary predicates in [20] is done via an ad hoc case analysis for 11 concrete predicates.…”

Section: Non-boolean Predicatesmentioning

confidence: 99%

“…Additive sparsifiability of such predicates is established in Proposition 14. Unlike in the multiplicative setting, it is not clear how to do this in a straightforward way similar to [20,14]. Instead, we associate with a given predicate P a vector v P in an appropriate vector space, identify special vectors that can be shown additively sparsifiable directly, show that linear combinations preserve sparsifiability, and argue that v P can be generated by the special vectors.…”

Section: Non-boolean Predicatesmentioning

confidence: 99%

See 1 more Smart Citation

Additive Sparsification of CSPs

Pelleg¹,

Živný²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The authors thank all reviewers of the extended abstract [6] and this full version of the paper for useful comments.…”

Section: Acknowledgementsmentioning

confidence: 99%