Document Open Access Logo

Computing Exact Minimum Cuts Without Knowing the Graph

Authors Aviad Rubinstein, Tselil Schramm, S. Matthew Weinberg

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2018.39.pdf
  • Filesize: 0.56 MB
  • 16 pages

Document Identifiers

Author Details

Aviad Rubinstein
Tselil Schramm
S. Matthew Weinberg

Cite AsGet BibTex

Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing Exact Minimum Cuts Without Knowing the Graph. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ITCS.2018.39

Abstract

We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query S \subset V, the oracle returns the size of the cut between S and V \ S. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with ~{O}(n^{5/3}) queries, and computing an exact global minimum cut of G with only ~{O}(n) queries (while learning the graph requires ~{\Theta}(n^2) queries).
Keywords
  • query complexity
  • minimum cut

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 459-467. SIAM, 2012. URL: http://portal.acm.org/citation.cfm?id=2095156&CFID=63838676&CFTOKEN=79617016.
  2. Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P. Woodruff, and Qin Zhang. On sketching quadratic forms. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 311-319. ACM, 2016. URL: http://dx.doi.org/10.1145/2840728.2840753.
  3. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Review, 56(2):315-334, 2014. URL: http://dx.doi.org/10.1137/130949117.
  4. András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015. URL: http://dx.doi.org/10.1137/070705970.
  5. Nader H. Bshouty and Hanna Mazzawi. Reconstructing weighted graphs with minimal query complexity. Theor. Comput. Sci., 412(19):1782-1790, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.12.055.
  6. Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. Subquadratic submodular function minimization. CoRR, abs/1610.09800, 2016. URL: http://arxiv.org/abs/1610.09800.
  7. Shiri Chechik, Yuval Emek, Boaz Patt-Shamir, and David Peleg. Sparse reliable graph backbones. Inf. Comput., 210:31-39, 2012. URL: http://dx.doi.org/10.1016/j.ic.2011.10.007.
  8. Sung-Soon Choi and Jeong Han Kim. Optimal query complexity bounds for finding graphs. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 749-758. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374484.
  9. Efim A. Dinitz, Alexander V. Karzanov, and Micael V. Lomonosov. On the structure of a family of minimum weighted cuts in a graph. Studies in Discrete Optimization, pages 290-306, 1976. Google Scholar
  10. Andrew Drucker, Fabian Kuhn, and Rotem Oshman. On the power of the congested clique model. In Magnús M. Halldórsson and Shlomi Dolev, editors, ACM Symposium on Principles of Distributed Computing, PODC '14, Paris, France, July 15-18, 2014, pages 367-376. ACM, 2014. URL: http://dx.doi.org/10.1145/2611462.2611493.
  11. Lester Randolph Ford Jr. and Delbert Ray Fulkerson. Flows in Networks. Princeton University Press, 1962. Google Scholar
  12. Satoru Fujishige. Submodular Functions and Optimization. Elsevier Science, 2005. Google Scholar
  13. Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. URL: http://dx.doi.org/10.1007/BF02579273.
  14. Nicholas J. A. Harvey. Matroid intersection, pointer chasing, and young’s seminormal representation of S_n. In Shang-Hua Teng, editor, Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 542-549. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347142.
  15. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.66.
  16. David R. Karger. Global min-cuts in rnc, and other ramifications of a simple min-cut algorithm. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas., pages 21-30. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313605.
  17. David R. Karger. Random sampling in cut, flow, and network design problems. Math. Oper. Res., 24(2):383-413, 1999. URL: http://dx.doi.org/10.1287/moor.24.2.383.
  18. David R. Karger and Matthew S. Levine. Fast augmenting paths by random sampling from residual graphs. SIAM J. Comput., 44(2):320-339, 2015. URL: http://dx.doi.org/10.1137/070705994.
  19. David R. Karger and Clifford Stein. A new approach to the minimum cut problem. J. ACM, 43(4):601-640, 1996. URL: http://dx.doi.org/10.1145/234533.234534.
  20. Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic global minimum cut of a simple graph in near-linear time. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 665-674. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746588.
  21. Dmitry Kogan and Robert Krauthgamer. Sketching cuts in graphs and hypergraphs. In Tim Roughgarden, editor, Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 367-376. ACM, 2015. URL: http://dx.doi.org/10.1145/2688073.2688093.
  22. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1049-1065. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.68.
  23. Hanna Mazzawi. Optimally reconstructing weighted graphs using queries. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 608-615. SIAM, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.51.
  24. Dalit Naor and Vijay V. Vazirani. Representing and enumerating edge connectivity cuts in RNC. In Frank K. H. A. Dehne, Jörg-Rüdiger Sack, and Nicola Santoro, editors, Algorithms and Data Structures, 2nd Workshop WADS '91, Ottawa, Canada, August 14-16, 1991, Proceedings, volume 519 of Lecture Notes in Computer Science, pages 273-285. Springer, 1991. URL: http://dx.doi.org/10.1007/BFb0028269.
  25. Robert Nishihara, Stefanie Jegelka, and Michael I. Jordan. On the convergence rate of decomposable submodular function minimization. In Zoubin Ghahramani, Max Welling, Corinna Cortes, Neil D. Lawrence, and Kilian Q. Weinberger, editors, Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pages 640-648, 2014. URL: http://papers.nips.cc/paper/5255-on-the-convergence-rate-of-decomposable-submodular-function-minimization.
  26. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed verification and hardness of distributed approximation. SIAM J. Comput., 41(5):1235-1265, 2012. URL: http://dx.doi.org/10.1137/11085178X.
  27. Peter Stobbe and Andreas Krause. Efficient minimization of decomposable submodular functions. In John D. Lafferty, Christopher K. I. Williams, John Shawe-Taylor, Richard S. Zemel, and Aron Culotta, editors, Advances in Neural Information Processing Systems 23: 24th Annual Conference on Neural Information Processing Systems 2010. Proceedings of a meeting held 6-9 December 2010, Vancouver, British Columbia, Canada., pages 2208-2216. Curran Associates, Inc., 2010. URL: http://papers.nips.cc/paper/4028-efficient-minimization-of-decomposable-submodular-functions.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact