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Algorithms - ESA 2015 pp 373–385Cite as

Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9294))

Abstract

Given an undirected, edge-weighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimum-weight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Relying on topological techniques, we provide a polynomial-time algorithm for this problem in the case where G is embedded on a fixed surface of genus g (e.g., when G is planar) and has a fixed number t of terminals. The running time is a polynomial of degree \(O\big(\sqrt{g^2+gt}\big)\) in the input size.

In the planar case, our result corrects an error in an extended abstract by Bentz [Int. Workshop on Parameterized and Exact Computation, 109–119, 2012]. The minimum multicut problem is also a generalization of the multiway cut problem, also known as the multiterminal cut problem; even for this special case, no dedicated algorithm was known for graphs embedded on surfaces.

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References

  1. Bateni, M., Hajiaghayi, M., Klein, P.N., Mathieu, C.: A polynomial-time approximation scheme for planar multiway cut. In: Proc. ACM-SIAM Symp. on Discrete Algorithms, pp. 639–655 (2012)

    Chapter  Google Scholar 

  2. Bentz, C.: On the complexity of the multicut problem in bounded tree-width graphs and digraphs. Disc. Applied Math. 156(10), 1908–1917 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bentz, C.: A simple algorithm for multicuts in planar graphs with outer terminals. Disc. Applied Math. 157, 1959–1964 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bentz, C.: A polynomial-time algorithm for planar multicuts with few source-sink pairs. In: International Workshop on Parameterized and Exact Computation, pp. 109–119 (2012), Also in arXiv:1206.3999

    Google Scholar 

  5. Chambers, E.W., Colin de Verdière, É., Erickson, J., Lazarus, F., Whittlesey, K.: Splitting (complicated) surfaces is hard. Comput. Geom.: Theory Appl. 41(1–2), 94–110 (2008)

    Google Scholar 

  6. Chambers, E.W., Erickson, J., Nayyeri, A.: Minimum cuts and shortest homologous cycles. In: Proc. Symp. on Computational Geometry, pp. 377–385. ACM (2009)

    Google Scholar 

  7. Chambers, E.W., Erickson, J., Nayyeri, A.: Homology flows, cohomology cuts. SIAM J. Comput. 41(6), 1605–1634 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colin de Verdière, É.: Topological algorithms for graphs on surfaces. Habilitation thesis, École normale supérieure (2012). http://www.di.ens.fr/~colin/

  9. Colin de Verdière, É., Erickson, J.: Tightening nonsimple paths and cycles on surfaces. SIAM J. Comput. 39(8), 3784–3813 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eppstein, D.: Dynamic generators of topologically embedded graphs. In: Proc. ACM-SIAM Symp. on Discrete Algorithms, pp. 599–608 (2003)

    Google Scholar 

  12. Erickson, J., Nayyeri, A.: Minimum cuts and shortest non-separating cycles via homology covers. In: Proc. ACM-SIAM Symp. on Discrete Algorithms, pp. 1166–1176 (2011)

    Google Scholar 

  13. Erickson, J., Nayyeri, A.: Shortest non-crossing walks in the plane. In: Proc. ACM-SIAM Symp. on Discrete Algorithms, pp. 297–308 (2011)

    Google Scholar 

  14. Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. J. Comput. System Sciences 55(1, part 1), 3–23 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Klein, P.N., Marx, D.: Solving planar k -terminal cut in \({O}(n^{c\sqrt{k}})\) time. In: Proc. Int. Coll. on Automata, Languages and Programming, vol. 1, pp. 569–580 (2012)

    Google Scholar 

  17. Marx, D.: A tight lower bound for planar multiway cut with fixed number of terminals. In: Proc. Int. Coll. on Automata, Languages and Programming, vol. 1, pp. 677–688 (2012)

    Google Scholar 

  18. Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Disc. Math. 12(1), 6–26 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)

    MATH  Google Scholar 

  20. Schrijver, A.: On the history of combinatorial optimization (till 1960). In: Handbook of Discrete Optimization, pp. 1–68. Elsevier (2005)

    Google Scholar 

  21. Stillwell, J.: Classical topology and combinatorial group theory, 2nd edn. Springer, New York (1993)

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Département d’informatique, École normale supérieure and CNRS, Paris, France

    Éric Colin de Verdière

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  1. Éric Colin de Verdière
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Correspondence to Éric Colin de Verdière .

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Editors and Affiliations

  1. University of Technology, Eindhoven, The Netherlands

    Nikhil Bansal

  2. Sapienza University of Rome, Rome, Italy

    Irene Finocchi

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de Verdière, É.C. (2015). Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_32

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  • DOI: https://doi.org/10.1007/978-3-662-48350-3_32

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  • Print ISBN: 978-3-662-48349-7

  • Online ISBN: 978-3-662-48350-3

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