Eden Pelleg
Stanislav Živný
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 : Dk→ { 0,1} of a fixed arity k on an arbitrary finite domain D.
- [1] . 2009. Graph sparsification in the semi-streaming model. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP’09), Part II(
Lecture Notes in Computer Science , Vol. 5556). Springer, 328–338.DOI: Google Scholar
Digital Library
- [2] . 1999. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28, 4 (1999), 1167–1181.
DOI: Google ScholarDigital Library - [3] . 1993. On sparse spanners of weighted graphs. Discrete & Computational Geometry 9, 1 (1993), 81–100.
DOI: Google ScholarDigital Library - [4] . 2016. On sketching quadratic forms. In Proceedings of the 7th ACM Conference on Innovations in Theoretical Computer Science (ITCS’16). ACM, 311–319.
DOI: Google ScholarDigital Library - [5] . 1985. Complexity of network synchronization. J. ACM 32, 4 (1985), 804–823.
DOI: Google ScholarDigital Library - [6] . 1998. New approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. SIAM J. Comput. 28, 1 (1998), 254–262.
DOI: Google ScholarDigital Library - [7] . 2019. New notions and constructions of sparsification for graphs and hypergraphs. In Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science (FOCS’19) (2019), 910–928.
DOI: Google ScholarCross Ref
- [8] . 2005. New constructions of (alpha, beta)-spanners and purely additive spanners. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’05). SIAM, 672–681. Retrieved from http://dl.acm.org/citation.cfm?id=1070432.1070526Google Scholar
- [9] . 2007. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithms 30, 4 (2007), 532–563.
DOI: Google ScholarCross Ref - [10] . 2014. Twice-ramanujan sparsifiers. SIAM Rev. 56, 2 (2014), 315–334.
DOI: Google ScholarDigital Library - [11] . 1996. Approximating s-t minimum cuts in \(\tilde{O}(n^2)\) time. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC’96). ACM, 47–55.
DOI: Google ScholarDigital Library - [12] . 2005. Sparse distance preservers and additive spanners. SIAM J. Discret. Math. 19, 4 (2005), 1029–1055.
DOI: Google ScholarDigital Library - [13] . 1980. Bigraphs versus digraphs via matrices. Journal of Graph Theory 4, 1 (1980), 51–73.
DOI: Google ScholarCross Ref - [14] . 2020. Sparsification of binary CSPs. SIAM J. Discret. Math. 34, 1 (2020), 825–842.
DOI: Google ScholarDigital Library - [15] . 2021. Vertex sparsification for edge connectivity. In Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’21). Retrieved from https://arxiv.org/abs/2007.07862Google ScholarCross Ref
- [16] . 2013. New additive spanners. In Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA’13). SIAM, 498–512.
DOI: Google ScholarCross Ref - [17] . 1998. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput. 28, 1 (1998), 210–236.
DOI: Google ScholarDigital Library - [18] . 2000. All-pairs almost shortest paths. SIAM J. Comput. 29, 5 (2000), 1740–1759.
DOI: Google ScholarDigital Library - [19] . 2020. Near-additive spanners and near-exact hopsets, a unified view. Bull. EATCS 130 (2020). Retrieved from https://dblp.org/rec/journals/eatcs/ElkinN20.html?view=bibtexGoogle Scholar
- [20] . 2017. Sparsification of two-variable valued constraint satisfaction problems. SIAM J. Discret. Math. 31, 2 (2017), 1263–1276.
DOI: Google ScholarDigital Library - [21] . 2019. A general framework for graph sparsification. SIAM J. Comput. 48, 4 (2019), 1196–1223. Google ScholarDigital Library
- [22] . 1998. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. J. Comput. Syst. Sci. 57, 3 (1998), 366–375.
DOI: Google ScholarDigital Library - [23] . 2015. Sketching cuts in graphs and hypergraphs. In Proceedings of the 6th ACM Conference on Innovations in Theoretical Computer Science (ITCS’15). ACM, 367–376.
DOI: Google ScholarDigital Library - [24] . 2016. Sparsified cholesky and multigrid solvers for connection laplacians. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC’16). ACM, 842–850.
DOI: Google ScholarDigital Library - [25] . 2010. Extensions and limits to vertex sparsification. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC’2010). ACM, 47–56.
DOI: Google ScholarDigital Library - [26] . 2009. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). IEEE Computer Society, 3–12.
DOI: Google ScholarDigital Library - [27] . 2013. On multiplicative lambda-approximations and some geometric applications. SIAM J. Comput. 42, 3 (2013), 855–883.
DOI: Google ScholarDigital Library - [28] . 1989. Graph spanners. Journal of Graph Theory 13, 1 (1989), 99–116.
DOI: Google ScholarCross Ref - [29] . 2021. Additive sparsification of CSPs. In Proceedings of the 29th Annual European Symposium on Algorithms (ESA’21)(
LIPIcs , Vol. 204). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 75:1–75:15.DOI: Google ScholarCross Ref - [30] . 2005. Deterministic constructions of approximate distance oracles and spanners. In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP’05)(
Lecture Notes in Computer Science , Vol. 3580). Springer, 261–272.DOI: Google ScholarDigital Library - [31] . 2019. Spectral sparsification of hypergraphs. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). 2570–2581.
DOI: Google ScholarCross Ref - [32] . 2011. Graph sparsification by effective resistances. SIAM J. Comput. 40, 6 (2011), 1913–1926.
DOI: Google ScholarDigital Library - [33] . 2004. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC’04). ACM, 81–90.
DOI: Google ScholarDigital Library - [34] . 2011. Spectral sparsification of graphs. SIAM J. Comput. 40, 4 (2011), 981–1025.
DOI: Google ScholarDigital Library - [35] . 2010. Additive spanners in nearly quadratic time. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP’10)(
Lecture Notes in Computer Science , Vol. 6198). Springer, 463–474.DOI: Google ScholarCross Ref
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- Additive Sparsification of CSPs
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