A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

  • Vincent Cohen-Addad
  • Arnaud de Mesmay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


Given a graph G cellularly embedded on a surface Σ of genus g, a cut graph is a subgraph of G such that cutting Σ along G yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any ε > 0, we show how to compute a (1 + ε) approximation of the shortest cut graph in time f(ε, g)n3.

Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Rué, Sau and Thilikos, which may be of independent interest.


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  1. 1.
    Bonsma, P.: Surface split decompositions and subgraph isomorphism in graphs on surfaces. In: Proc. of the Symp. on Theoretical Aspects of Computer Science, STACS 2012, pp. 531–542 (2012)Google Scholar
  2. 2.
    Borradaile, G., Demaine, E.D., Tazari, S.: Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica 68(2), 287–311 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borradaile, G., Klein, P.N., Mathieu, C.: An O(nlogn) approximation scheme for Steiner tree in planar graphs. ACM Trans. on Algorithms 5(3) (2009)Google Scholar
  4. 4.
    Cohen-Addad, V., de Mesmay, A.: A fixed paramater tractable approximation scheme for the optimal cut graph of a surface (2015) (in preparation)Google Scholar
  5. 5.
    Demaine, E.D., Hajiaghayi, M., Kawarabayashi, K.I.: Contraction decomposition in h-minor-free graphs and algorithmic applications. In: Proc. of the ACM Symp. on Theory of Computing, STOC 2011, pp. 441–450. ACM (2011)Google Scholar
  6. 6.
    Demaine, E.D., Hajiaghayi, M., Mohar, B.: Approximation algorithms via contraction decomposition. Combinatorica, 533–552 (2010)Google Scholar
  7. 7.
    Erickson, J.: Combinatorial optimization of cycles and bases. In: Zomorodian, A. (ed.) Computational Topology. Proc. of Symp. in Applied Mathematics, AMS (2012)Google Scholar
  8. 8.
    Erickson, J., Har-Peled, S.: Optimally cutting a surface into a disk. Discrete & Computational Geometry 31(1), 37–59 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erickson, R.E., Monma, C.L., Veinott Jr., A.F.: Send-and-split method for minimum-concave-cost network flows. Mathematics of Operations Research 12(4), 634–664 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fomin, F.V., Thilikos, D.M.: On self duality of pathwidth in polyhedral graph embeddings. Journal of Graph Theory 55(1), 42–54 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Inkmann, T.: Tree-based decompositions of graphs on surfaces and applications to the Traveling Salesman Problem. Ph.D. thesis, Georgia Inst. of Technology (2007)Google Scholar
  12. 12.
    Marx, D.: Parameterized complexity and approximation algorithms. The Computer Journal 51(1), 60–78 (2008)CrossRefGoogle Scholar
  13. 13.
    Mazoit, F.: Tree-width of hypergraphs and surface duality. J. Comb. Theory, Ser. B 102(3), 671–687 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mohar, B., Thomassen, C.: Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press (2001)Google Scholar
  15. 15.
    Pilipczuk, M., Pilipczuk, M., Sankowski, P., van Leeuwen, E.J.: Network sparsification for steiner problems on planar and bounded-genus graphs. In: Proceedings of Foundations of Computer Science (FOCS), pp. 276–285 (2014)Google Scholar
  16. 16.
    Robertson, N., Seymour, P.: Graph minors. X. obstructions to tree-decomposition. J. Combin. Theory Ser. B 52(2), 153–190 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rué, J., Sau, I., Thilikos, D.M.: Dynamic programming for graphs on surfaces. ACM Trans. Algorithms 10(2), 8:1–8:26 (2014)Google Scholar
  18. 18.
    Thilikos, D.M., Serna, M., Bodlaender, H.L.: Constructive linear time algorithms for small cutwidth and carving-width. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 192–203. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Colin de Verdière, É.: Topological algorithms for graphs on surfaces (2012), habilitation thesis.
  20. 20.
    Wood, Z., Hoppe, H., Desbrun, M., Schröder, P.: Removing excess topology from isosurfaces. ACM Transactions on Graphics 23(2), 190–208 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Vincent Cohen-Addad
    • 1
  • Arnaud de Mesmay
    • 2
  1. 1.Département d’informatiqueÉcole normale supérieureParisFrance
  2. 2.IST AustriaViennaAustria

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