Solving the 2-Disjoint Connected Subgraphs Problem Faster Than 2n

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z 1 ,Z 2 , asks whether it is possible to find disjoint sets A 1 ,A 2 , such that Z 1 ⊆ A 1 , Z 2 ⊆ A 2 and A 1 ,A 2 induce connected subgraphs. While the naive algorithm runs in O(2 n n O(1)) time, solutions with complexity of form O((2 − ε) n ) have been found only for special graph classes [15, 19]. In this paper we present an O(1.933 n ) algorithm for 2-Disjoint Connected Subgraphs in general case, thus breaking the 2 n barrier. As a counterpoise of this result we show that if we parameterize the problem by the number of non-terminal vertices, it is hard both to speed up the brute-force approach and to find a polynomial kernel.


Polynomial Kernel Reduction Rule Valid Solution Articulation Point Subgraph Problem 
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  1. 1.
    Björklund, A.: Determinant sums for undirected hamiltonicity. In: 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 173–182 (2010)Google Scholar
  2. 2.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution. In: 39th Annual ACM Symposium on Theory of Computing (STOC), pp. 67–74 (2007)Google Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Thomasse, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels, technical Report UU-CS-2008-030, Institute of Information and Computing Sciences, Utrecht University, Netherlands (2008)Google Scholar
  5. 5.
    Calabro, C., Impagliazzo, R., Paturi, R.: The Complexity of Satisfiability of Small Depth Circuits. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 75–85. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNF-SAT. CoRR abs/1112.2275 (2011)Google Scholar
  7. 7.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: 52th Annual IEEE Symposium on Foundations of Computer Science, FOCS (2011) (to appear)Google Scholar
  8. 8.
    Cygan, M., Pilipczuk, M.: Exact and approximate bandwidth. Theor. Comput. Sci. 411(40-42), 3701–3713 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Scheduling Partially Ordered Jobs Faster Than 2n. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 299–310. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through Colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer (2010)Google Scholar
  12. 12.
    Fomin, F.V., Todinca, I., Villanger, Y.: Exact Algorithm for the Maximum Induced Planar Subgraph Problem. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 287–298. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56(5), 1–32 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gray, C., Kammer, F., Löffler, M., Silveira, R.I.: Removing Local Extrema from Imprecise Terrains. CoRR abs/1002.2580 (2010)Google Scholar
  15. 15.
    van ’t Hof, P., Paulusma, D., Woeginger, G.J.: Partitioning graphs into connected parts. Theor. Comput. Sci. 410(47-49), 4834–4843 (2009)zbMATHCrossRefGoogle Scholar
  16. 16.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lokshtanov, D., Marx, D., Saurabh, S.: Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 777–789 (2011)Google Scholar
  18. 18.
    Patrascu, M., Williams, R.: On the possibility of faster SAT algorithms. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1065–1075 (2010)Google Scholar
  19. 19.
    Paulusma, D., van Rooij, J.M.M.: On partitioning a graph into two connected subgraphs. Theor. Comput. Sci. 412(48), 6761–6769 (2011)zbMATHCrossRefGoogle Scholar
  20. 20.
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63(1), 65–110 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/Exclusion Meets Measure and Conquer. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 554–565. Springer, Heidelberg (2009)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 1
  • Michał Pilipczuk
    • 2
  • Jakub Onufry Wojtaszczyk
    • 3
  1. 1.Institute of InformaticsUniversity of WarsawPoland
  2. 2.Department of InformaticsUniversity of BergenNorway
  3. 3.Google Inc.WarsawPoland

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