Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

Global Minimum Cuts in Surface-Embedded Graphs

  • Erin W. Chambers
  • Jeff Erickson
  • Kyle Fox
  • Amir Nayyeri
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_683-1

Years and Authors of Summarized Original Work

2009; Chambers, Erickson, Nayyeri

2011; Erickson, Nayyeri

2012; Erickson, Fox, Nayyeri

Problem Definition

Given a graph G in which every edge has a nonnegative capacity, the goal of the minimum-cut problem is to find a subset of edges of G with minimum total capacity whose deletion disconnects G. The closely related minimum (s, t)-cut problem further requires two specific vertices s and t to be separated by the deleted edges. Minimum cuts and their generalizations play a central role in divide-and-conquer and network optimization algorithms.

The fastest algorithms known for computing minimum cuts in arbitrary graphs run in roughly O(mn) time for graphs with n vertices and medges. However, even faster algorithms are known for graphs with additional topological structure. This entry sketches algorithms to compute minimum cuts in near-linear time when the input graph can be drawn on a surface with bounded genus – informally, a sphere with a...


Topological graph theory Graph embedding Minimum cuts Homology Covering spaces Fixed-parameter tractability 
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Recommended Reading

  1. 1.
    Borradaile G, Klein P (2009) An O(nlogn) algorithm for maximum st-flow in a directed planar graph. J ACM 56(2):9:1–30Google Scholar
  2. 2.
    Cabello S (2010) Finding shortest contractible and shortest separating cycles in embedded graphs. ACM Trans Algorithms 6(2):24:1–24:18Google Scholar
  3. 3.
    Cabello S, Chambers EW, Erickson J (2013) Multiple-source shortest paths in embedded graphs. SIAM J Comput 42(4):1542–1571CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chambers E, Eppstein D (2013) Flows in one-crossing-minor-free graphs. J Graph Algorithms Appl 17(3):201–220. doi:10.7155/jgaa.00291CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chambers EW, Colin de Verdière É, Erickson J, Lazarus F, Whittlesey K (2008) Splitting (complicated) surfaces is hard. Comput Geom Theory Appl 41(1–2):94–110CrossRefzbMATHGoogle Scholar
  6. 6.
    Chambers EW, Erickson J, Nayyeri A (2009) Minimum cuts and shortest homologous cycles. In: Proceedings of the 25th annual symposium computational geometry, Aarhus, pp 377–385Google Scholar
  7. 7.
    Chambers EW, Erickson J, Nayyeri A (2012) Homology flows, cohomology cuts. SIAM J Comput 41(6):1605–1634CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Erickson J (2012) Combinatorial optimization of cycles and bases. In: Zomorodian A (ed) Advances in applied and computational topology. Invited survey for an AMS short course on computational topology at the 2011 joint mathematics meetings, New Orleans. Proceedings of symposia in applied mathematics, vol 70. American Mathematical Society, pp 195–228Google Scholar
  9. 9.
    Erickson J, Nayyeri A (2011) Minimum cuts and shortest non-separating cycles via homology covers. In: Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, San Francisco, pp 1166–1176Google Scholar
  10. 10.
    Hagerup T, Katajainen J, Nishimura N, Ragde P (1998) Characterizing multiterminal flow networks and computing flows in networks of small treewidth. J Comput Syst Sci 57(3):366–375CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Itai A, Shiloach Y (1979) Maximum flow in planar networks. SIAM J Comput 8:135–150CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Italiano GF, Nussbaum Y, Sankowski P, Wulff-Nilsen C (2011) Improved algorithms for min cut and max flow in undirected planar graphs. In: Proceedings of the 43rd annual ACM symposium theory of computing, San Jose, pp 313–322Google Scholar
  13. 13.
    Kutz M (2006) Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time. In: Proceedings of the 22nd annual symposium on computational geometry, Sedona, pp 430–438Google Scholar
  14. 14.
    Łącki J, Sankowski P (2011) Min-cuts and shortest cycles in planar graphs in O(nloglogn) time. In: Proceedings of the 19th annual European symposium on algorithms, Saarbrücken. Lecture notes in computer science, vol 6942. Springer, pp 155–166Google Scholar
  15. 15.
    Reif J (1983) Minimum s-t cut of a planar undirected network in O(nlog2 n) time. SIAM J Comput 12:71–81CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Erin W. Chambers
    • 1
  • Jeff Erickson
    • 2
  • Kyle Fox
    • 3
  • Amir Nayyeri
    • 4
  1. 1.Department of Computer Science and MathematicsSaint Louis UniversitySt. Louis, MOUSA
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbana, ILUSA
  3. 3.Institute for Computational and Experimental Research in MathematicsBrown UniversityProvidence, RIUSA
  4. 4.Department of Electrical Engineering and Computer ScienceOregon State UniversityCorvallis, ORUSA