Advertisement

On the Parameterized Complexity of Computing Graph Bisections

  • René van Bevern
  • Andreas Emil Feldmann
  • Manuel Sorge
  • Ondřej Suchý
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)

Abstract

The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem.

We show that Bisection is FPT w.r.t. the minimum cut size if there is an optimum bisection that cuts into a given constant number of connected components. Our algorithm applies to the more general Balanced Biseparator problem where vertices need to be removed instead of edges. We prove that this problem is W[1]-hard w.r.t. the minimum cut size and the number of cut out components.

For Bisection we further show that no polynomial-size kernels exist for the cut size parameter. In fact, we show this for all parameters that are polynomial in the input size and that do not increase when taking disjoint unions of graphs. We prove fixed-parameter tractability for the distance to constant cliquewidth if we are given the deletion set. This implies fixed-parameter algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set.

Keywords

Vertex Cover Vertex Weight Problem Kernel Bisection Width Bisection Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andreev, K., Räcke, H.: Balanced graph partitioning. Theory of Computing Systems 39(6), 929–939 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arbenz, P.: Personal communication, ETH Zürich (2013)Google Scholar
  3. 3.
    P. Arbenz, G. van Lenthe, U. Mennel, R. Müller, and M. Sala. Multi-level μ-finite element analysis for human bone structures. In Proc. 8th PARA, volume 4699 of LNCS, pages 240–250. Springer, 2007.CrossRefGoogle Scholar
  4. 4.
    Bhatt, S.N., Leighton, F.T.: A framework for solving VLSI graph layout problems. J. Comput. Syst. Sci. 28(2), 300–343 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Science 209(1-2), 1–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proc. 28th STACS. LIPIcs, vol. 9, pp. 165–176. Dagstuhl (2011)Google Scholar
  8. 8.
    Bui, T.N., Peck, A.: Partitioning planar graphs. SIAM J. Comput. 21(2), 203–215 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bui, T.N., Chaudhuri, S., Leighton, F.T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7(2), 171–191 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1-3), 77–114 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.F.: Customizable route planning. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 376–387. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Diestel, R.: Graph Theory, 4th edn. Graduate Texts in Mathematics, vol. 173. Springer (2010)Google Scholar
  14. 14.
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Feldmann, A.E.: Fast balanced partitioning is hard, even on grids and trees. Theor. Comput. Sci. 485, 61–68 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Feldmann, A.E., Foschini, L.: Balanced partitions of trees and applications. In: Proc. 29th STACS. LIPIcs, vol. 14, pp. 100–111. Dagstuhl (2012)Google Scholar
  17. 17.
    Feldmann, A.E., Widmayer, P.: An \(\mathcal{O}(n^4)\) time algorithm to compute the bisection width of solid grid graphs. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 143–154. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar \(\mathcal{F}\)-deletion: Approximation, kernelization and optimal fpt algorithms. In: Proc. 53rd FOCS, pp. 470–479. IEEE Computer Society (2012)Google Scholar
  19. 19.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co. (1979)Google Scholar
  20. 20.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Science 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  22. 22.
    Hliněný, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008)CrossRefGoogle Scholar
  23. 23.
    Khot, S.A., Vishnoi, N.K.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: Proc. 46th FOCS, pp. 53–62. IEEE Computer Society (2005)Google Scholar
  24. 24.
    Kloks, T., Lee, C.M., Liu, J.: New algorithms for k-face cover, k-feedback vertex set, and k-disjoint cycles on plane and planar graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 282–295. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  25. 25.
    Kwatra, V., Schödl, A., Essa, I., Turk, G., Bobick, A.: Graphcut textures: Image and video synthesis using graph cuts. ACM T. Graphic. 22(3), 277–286 (2003)CrossRefGoogle Scholar
  26. 26.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    MacGregor, R.M.: On Partitioning a Graph: a Theoretical and Empirical Study. PhD thesis, University of California, Berkeley (1978)Google Scholar
  28. 28.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Marx, D., O’Sullivan, B., Razgon, I.: Treewidth reduction for constrained separation and bipartization problems. In: Proc. 27th STACS. LIPIcs, vol. 5, pp. 561–572. Dagstuhl (2010)Google Scholar
  30. 30.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. CoRR, abs/1110.4765 (2011)Google Scholar
  31. 31.
    Oum, S.: Approximating rank-width and clique-width quickly. ACM T. Algorithms 5 (1) (2008)Google Scholar
  32. 32.
    Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proc. 40th STOC, pp. 255–264. ACM (2008)Google Scholar
  33. 33.
    Soumyanath, K., Deogun, J.S.: On the bisection width of partial k-trees. In: Proc. 20th Southeastern Conference on Combinatorics, Graph Theory, and Computing. Congressus Numerantium, vol. 74, pp. 25–37 (1990)Google Scholar
  34. 34.
    Wiegers, M.: The k-section of treewidth restricted graphs. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 530–537. Springer, Heidelberg (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • René van Bevern
    • 1
  • Andreas Emil Feldmann
    • 2
  • Manuel Sorge
    • 1
  • Ondřej Suchý
    • 3
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany
  2. 2.Combinatorics & OptimizationUniversity of WaterlooCanada
  3. 3.Faculty of Information TechnologyCzech Technical University in PragueCzech Republic

Personalised recommendations