Additive Sparsification of CSPs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866910285781008384 |
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| author | Pelleg, Eden Živný, Stanislav |
| author_facet | Pelleg, Eden Živný, Stanislav |
| contents | 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 $P:\{0,1\}^k\to\{0,1\}$ 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 $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_14757 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Additive Sparsification of CSPs Pelleg, Eden Živný, Stanislav Data Structures and Algorithms Discrete Mathematics 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 $P:\{0,1\}^k\to\{0,1\}$ 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 $D$. |
| title | Additive Sparsification of CSPs |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2106.14757 |