More Web Proxy on the site http://driver.im/

research-article

Sketching Cuts in Graphs and Hypergraphs

Authors:

Robert KrauthgamerAuthors Info & Claims

ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science

Pages 367 - 376

https://doi.org/10.1145/2688073.2688093

Published: 11 January 2015 Publication History

Abstract

Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that (1--ε)-approximation for Max-Cut must use n^{1-O(ε)} space; moreover, beating 4/5-approximation requires polynomial space. For the sketching model, we show that every r-uniform hypergraph admits a (1+ ε)-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(ε^-2n(r+log n)) edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems).

References

[1]

K. J. Ahn and S. Guha. Graph sparsification in the semi-streaming model. In 36th International Colloquium on Automata, Languages and Programming: Part II, ICALP '09, pages 328--338. Springer-Verlag, 2009. arXiv:0902:0140.

Digital Library

[2]

K. J. Ahn, S. Guha, and A. McGregor. Analyzing graph structure via linear measurements. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, pages 459--467. SIAM, 2012.

Digital Library

[3]

K. J. Ahn, S. Guha, and A. McGregor. Graph sketches: Sparsification, spanners, and subgraphs. In Proceedings of the 31st Symposium on Principles of Database Systems, PODS '12, pages 5--14. ACM, 2012.

Digital Library

[4]

A. Andoni, R. Krauthgamer, and D. P. Woodru. The sketching complexity of graph cuts. CoRR, abs/1403.7058, 2014. arXiv:1403:7058.

[5]

A. A. Bencz ur and D. R. Karger. Approximating s-t minimum cuts in Õ (n²) time. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC '96, pages 47--55, New York, NY, USA, 1996. ACM.

Digital Library

[6]

A. A. Bencz ur and D. R. Karger. Randomized approximation schemes for cuts and ows in capacitated graphs. CoRR, cs.DS/0207078, 2002. arXiv:cs/0207078.

[7]

M. K. de Carli Silva, N. J. A. Harvey, and C. M. Sato. Sparse sums of positive semidefinite matrices. CoRR, abs/1107.0088, 2011.arXiv:1107:0088.

[8]

H. Dell and D. van Melkebeek. Satisability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC '10, pages 251--260. ACM, 2010.

Digital Library

[9]

E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov. On the structure of the system of minimum edge cuts in a graph. Issledovaniya po Diskretnoi Optimizatsii, pages 290--306, 1976. URL: http://alexander karzanov:net/ScannedOld/76_cactus_transl:pdf.

[10]

A. Goel, M. Kapralov, and I. Post. Single pass sparsification in the streaming model with edge deletions. CoRR, abs/1203.4900, 2012.arXiv:1203:4900.

[11]

M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisability problems using semidefinite programming. J. ACM, 42(6):1115--1145, Nov. 1995.

Digital Library

[12]

R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9:551--570, 1961.

[13]

M. Henzinger and D. P. Williamson. On the number of small cuts in a graph. Information Processing Letters, 59(1):41--44, 1996.

Digital Library

[14]

P. Indyk, A. McGregor, I. Newman, and K. Onak. Open questions in data streams, property testing, and related topics. http://people:cs:umass:edu/~mcgregor/papers/11-openproblems:pdf, 2011. See Also http://sublinear:info/45.

[15]

M. Kapralov, S. Khanna, and M. Sudan. Streaming lower bounds for approximating MAX-CUT. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, 2015. To Appear. arXiv:1409:2138.

Digital Library

[16]

M. Kapralov, Y. T. Lee, C. Musco, C. Musco, and A. Sidford. Single pass spectral sparsification in dynamic streams. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science, 2014. arXiv:1407:1289.

Digital Library

[17]

D. R. Karger. Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '93, pages 21{30, Philadelphia, PA, USA, 1993. Society for Industrial and Applied Mathematics. URL: http://dl:acm:org/citation:cfm?id=313559:313605.

Digital Library

[18]

D. R. Karger. Better random sampling algorithms for flows in undirected graphs. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '98, pages 490--499. SIAM, 1998. URL: http://dl:acm:org/citation:cfm?id=314613:314833.

Digital Library

[19]

D. R. Karger. Random sampling in cut, ow, and network design problems. Mathematics of Operations Research, 24(2):383--413, 1999.

[20]

T. Kaufman, D. Kazhdan, and A. Lubotzky. Ramanujan complexes and bounded degree topological expanders. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science, 2014. arXiv:1409:1397.

Digital Library

[21]

S. Khot, G. Kindler, E. Mossel, and R. O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319--357, 2007.

Digital Library

[22]

R. Klimmek and F. Wagner. A simple hypergraph min cut algorithm. Technical Report B 96-02, Freie Universität Berlin, Fachbereich Mathematik, 1996. URL: http://edocs:fuberlin:de/docs/servlets/MCRFileNodeServlet/FUDOCS_derivate_00000000297/1996_02:pdf.

[23]

F. T. Leighton and A. Moitra. Extensions and limits to vertex sparsification. In 42nd ACM symposium on Theory of computing, STOC, pages 47--56. ACM, 2010.

Digital Library

[24]

M. V. Lomonosov and V. Polesskii. Lower bound of network reliability. Problemy Peredachi Informatsii, 8(2):47--53, 1972. URL: http://www:mathnet:ru/links/36bd620cb75111781cef454d72f0d773/ppi824:pdf.

[25]

A. Louis. Hypergraph Markov operators, eigenvalues and approximation algorithms. CoRR, abs/1408.2425,2014.arXiv:1408:2425.

[26]

A. Louis and Y. Makarychev. Approximation algorithms for hypergraph small set expansion and small set vertex expansion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), volume 28, pages 339--355, 2014.

[27]

A. McGregor. Graph stream algorithms: A survey. SIGMOD Rec., 43(1):9--20, May 2014.

Digital Library

[28]

A. Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In 50th Annual Symposium on Foundations of Computer Science, FOCS, pages 3--12. IEEE, 2009.

Digital Library

[29]

H. Nagamochi, K. Nishimura, and T. Ibaraki. Computing all small cuts in an undirected network. SIAM Journal on Discrete Mathematics, 10(3):469--481, 1997.

Digital Library

[30]

I. Newman and Y. Rabinovich. On multiplicative-approximations and some geometric applications. SIAM Journal on Computing, 42(3):855--883, 2013.

[31]

D. Peleg and A. A. Schäer. Graph spanners. J. Graph Theory, 13(1):99--116, 1989.

[32]

M. Thorup and U. Zwick. Approximate distance oracles. J. ACM, 52(1):1--24, 2005.

Digital Library

[33]

Y. Yamaguchi, A. Ogawa, A. Takeda, and S. Iwata. Cyber security analysis of power networks by hypergraph cut algorithms. In Proceedings of the Fifth Annual IEEE International Conference on Smart Grid Communications, 2014. To appear. URL: http://www:keisu:t:u-tokyo:ac:jp/research/techrep/data/2014/METR14--12:pdf.

[34]

W. Yu and E. Verbin. The streaming complexity of cycle counting, sorting by reversals, and other problems. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 11--25, 2011.

Digital Library

[35]

M. Zelke. Algorithms for Streaming Graphs. PhD thesis, Mathematisch-Naturwissenschaftliche Fakultät II, Humboldt-Universität zu Berlin, 2009. Published at Südwestdeutscher Verlag für Hochschulschriften. URL: http://www:tks:informatik:uni-frankfurt:de/data/doc/diss:pdf.

Digital Library

[36]

M. Zelke. Intractability of min- and max-cut in streaming graphs. Inf. Process. Lett., 111(3):145--150, Jan. 2011.

Digital Library

[37]

J. Zhang. A survey on streaming algorithms for massive graphs. In C. C. Aggarwal and H. Wang, editors, Managing and Mining Graph Data, volume 40 of Advances in Database Systems, pages 393--420. Springer, 2010.

Cited By

Oliver PZhang EZhang Y(2024)Scalable Hypergraph VisualizationIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332659930:1(595-605)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3326599
Khanna SPutterman ASudan M(2024)Near-Optimal Size Linear Sketches for Hypergraph Cut Sparsifiers2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00105(1669-1706)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00105
Singer NSudan MVelusamy S(2024)Streaming approximation resistance of every ordering CSPcomputational complexity10.1007/s00037-024-00252-533:1Online publication date: 29-May-2024
https://doi.org/10.1007/s00037-024-00252-5
Show More Cited By

Index Terms

Sketching Cuts in Graphs and Hypergraphs

Recommendations

Strong Transversals in Hypergraphs and Double Total Domination in Graphs

Let $H$ be a 3-uniform hypergraph of order $n$ and size $m$, and let $T$ be a subset of vertices of $H$. The set $T$ is a strong transversal in $H$ if $T$ contains at least two vertices from every edge of $H$. The strong transversal number $\tau_s(H)$ ...
Erdős-Hajnal-type theorems in hypergraphs

The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^@d^(^H^), where @d(H)>0 depends only on the ...
On 2-Colorings of Hypergraphs

In this article, we continue the study of 2-colorings in hypergraphs. A hypergraph is 2-colorable if there is a 2-coloring of the vertices with no monochromatic hyperedge. Let Hk denote the class of all k-uniform k-regular hypergraphs. It is known see ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Conferences

ITCS '15: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science

January 2015

404 pages

ISBN:9781450333337

DOI:10.1145/2688073

Program Chair:
Tim Roughgarden
Stanford University

Copyright © 2015 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 January 2015

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

USA-Israel BSF
Citi Foundation
Israel Science Foundation

Conference

ITCS'15

Sponsor:

SIGACT

ITCS'15: Innovations in Theoretical Computer Science

January 11 - 13, 2015

Rehovot, Israel

Acceptance Rates

ITCS '15 Paper Acceptance Rate 45 of 159 submissions, 28%;

Overall Acceptance Rate 172 of 513 submissions, 34%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

40
Total Citations
View Citations
319
Total Downloads

Downloads (Last 12 months)36
Downloads (Last 6 weeks)4

Reflects downloads up to 18 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Oliver PZhang EZhang Y(2024)Scalable Hypergraph VisualizationIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332659930:1(595-605)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3326599
Khanna SPutterman ASudan M(2024)Near-Optimal Size Linear Sketches for Hypergraph Cut Sparsifiers2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00105(1669-1706)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00105
Singer NSudan MVelusamy S(2024)Streaming approximation resistance of every ordering CSPcomputational complexity10.1007/s00037-024-00252-533:1Online publication date: 29-May-2024
https://doi.org/10.1007/s00037-024-00252-5
Pelleg EŽivný S(2023)Additive Sparsification of CSPsACM Transactions on Algorithms10.1145/362582420:1(1-18)Online publication date: 13-Nov-2023
https://dl.acm.org/doi/10.1145/3625824
Assadi S(2023)Recent Advances in Multi-Pass Graph Streaming Lower BoundsACM SIGACT News10.1145/3623800.362380854:3(48-75)Online publication date: 11-Sep-2023
https://dl.acm.org/doi/10.1145/3623800.3623808
Fox KPanigrahi DZhang F(2023)Minimum Cut and Minimum k-Cut in Hypergraphs via Branching ContractionsACM Transactions on Algorithms10.1145/357016219:2(1-22)Online publication date: 15-Apr-2023
https://dl.acm.org/doi/10.1145/3570162
Assadi SSundaresan JSaha BServedio R(2023)(Noisy) Gap Cycle Counting Strikes Back: Random Order Streaming Lower Bounds for Connected Components and BeyondProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585192(183-195)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585192
Jambulapati ALiu YSidford ASaha BServedio R(2023)Chaining, Group Leverage Score Overestimates, and Fast Spectral Hypergraph SparsificationProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585136(196-206)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585136
Saxena RSinger NSudan MVelusamy S(2023)Improved Streaming Algorithms for Maximum Directed Cut via Smoothed Snapshots2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00055(855-870)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00055
Kallaugher JParekh O(2022)The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00054(498-506)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00054
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents