Document Open Access Logo

All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

Authors Glencora Borradaile, David Eppstein, Amir Nayyeri, Christian Wulff-Nilsen

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2016.22.pdf
  • Filesize: 0.56 MB
  • 16 pages

Document Identifiers

Subject Classification

Keywords
  • minimum cuts
  • surface-embedded graphs
  • Gomory-Hu tree

Metrics

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

Abstract

For an undirected n-vertex graph G with non-negative edge-weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a minimum st-cut in G? We solve this problem in preprocessing time O(n log^3 n) for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space O(n) that can answer minimum-cut queries in constant time.  The dependence on the genus of the input graph in our preprocessing time is 2^{O(g^2)}.

Cite As Get BibTex

Glencora Borradaile, David Eppstein, Amir Nayyeri, and Christian Wulff-Nilsen. All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.22

Author Details

Glencora Borradaile
David Eppstein
Amir Nayyeri
Christian Wulff-Nilsen

References

  1. G. Borradaile, E. Chambers, K. Fox, and A. Nayyeri. Computing minimum homology basis and minimum cycle basis in surface embedded graphs. In submission., 2015. Google Scholar
  2. G. Borradaile, P.N. Klein, S. Mozes, Y. Nussbaum, and C. Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. In Proc. 52nd Symp. Found. of Computer Science (FOCS 2011), pages 170-179, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.73.
  3. G. Borradaile, P. Sankowski, and C. Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. In Proc. 51st IEEE Symp. Foundations of Computer Science (FOCS 2010), pages 601-610, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.63.
  4. G. Borradaile, P. Sankowski, and C. Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. ACM Trans. Algorithms, 2014. To appear. Google Scholar
  5. S. Cabello, É. Colin de Verdière, and F. Lazarus. Finding shortest non-trivial cycles in directed graphs on surfaces. In Proc. of the 26th Annual Symposium on Computational Geometry, SoCG'10, pages 156-165, New York, NY, USA, 2010. ACM. URL: http://dx.doi.org/10.1145/1810959.1810988.
  6. E. Chambers, J. Erickson, and A. Nayyeri. Homology flows, cohomology cuts. In Proc. 41st ACM Symposium on Theory of Computing (STOC'09), pages 273-282, 2009. URL: http://dx.doi.org/10.1145/1536414.1536453.
  7. E. Chambers, J. Erickson, and A. Nayyeri. Minimum cuts and shortest homologous cycles. In Proc. 25th Symp. Computational Geometry (SoCG'09), pages 377-385, 2009. URL: http://dx.doi.org/10.1145/1542362.1542426.
  8. Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Homology flows, cohomology cuts. SIAM J. Comput., 41:1605-1634, 2009. URL: http://dx.doi.org/10.1137/090766863.
  9. E. Demaine, G. Landau, and O. Weimann. On Cartesian trees and range minimum queries. Algorithmica, 68(3):610-625, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9683-x.
  10. J. Erickson. Shortest non-trivial cycles in directed surface graphs. In Proc. of the 27th Annual Symp. on Computational Geometry, SoCG'11, pages 236-243, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1998196.1998231.
  11. J. Erickson, K. Fox, and A. Nayyeri. Global minimum cuts in surface embedded graphs. In Proc. 23rd Annual ACM-SIAM Symp. on Discrete Algorithms, pages 1309-1318, 2012. Google Scholar
  12. J. Erickson and A. Nayyeri. Minimum cuts and shortest non-separating cycles via homology covers. In Proc. 22nd ACM-SIAM Symp. Discrete Algorithms (SODA 2011), pages 1166-1176, 2011. Google Scholar
  13. J. Erickson and A. Sidiropoulos. A near-optimal approximation algorithm for asymmetric tsp on embedded graphs. In Proc. 30th Annual Symposium on Computational Geometry, SOCG'14, pages 130:130-130:135. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582136.
  14. J. Fakcharoenphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci., 72(5):868-889, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2005.05.007.
  15. R. Gomory and T. Hu. Multi-terminal network flows. Journal of SIAM, 9(4):551-570, 1961. URL: http://dx.doi.org/10.1137/0109047.
  16. D. Gusfield. Very simple methods for all pairs network flow analysis. SIAM J. Comput., 19(1):143-155, 1990. URL: http://dx.doi.org/10.1137/0219009.
  17. D. Hartvigsen and R. Mardon. The all-pairs min cut problem and the minimum cycle basis problem on planar graphs. SIAM J. on Discrete Math., 7(3):403-418, 1994. URL: http://dx.doi.org/10.1137/S0895480190177042.
  18. A. Hatcher. Algebraic Topology. Cambridge University Press, 2002. Google Scholar
  19. J. Łacki, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen. Single source - all sinks max flows in planar digraphs. In Proc. 53rd Symp. Found. of Computer Science (FOCS 2012), pages 599-608, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.66.
  20. G. Miller. Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci., 32(3):265-279, 1986. URL: http://dx.doi.org/10.1016/0022-0000(86)90030-9.
  21. K. Mulmuley, V. Vazirani, and U. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):345-354, 1987. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact