Advertisement

Algorithms for Cut Problems on Trees

  • Iyad Kanj
  • Guohui Lin
  • Tian Liu
  • Weitian Tong
  • Ge Xia
  • Jinhui Xu
  • Boting Yang
  • Fenghui Zhang
  • Peng Zhang
  • Binhai Zhu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

Abstract

We study the multicut on trees and the generalized multiway cut on trees problems. For the multicut on trees problem, we present a parameterized algorithm that runs in time \(O^{*}(\rho ^k)\), where \(\rho = \sqrt{\sqrt{2} + 1} \approx 1.555\) is the positive root of the polynomial \(x^4-2x^2-1\). This improves the current-best algorithm of Chen et al. that runs in time \(O^{*}(1.619^k)\). By reducing generalized multiway cut on trees to multicut on trees, our results give a parameterized algorithm that solves the generalized multiway cut on trees problem in time \(O^{*}(\rho ^k)\). We also show that the generalized multiway cut on trees problem is solvable in polynomial time if the number of terminal sets is a fixed constant.

References

  1. 1.
    Avidor, A., Langberg, M.: The multi-multiway cut problem. Theor. Comput. Sci. 377(1–3), 35–42 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: STOC, pp. 459–468 (2011)Google Scholar
  3. 3.
    Bousquet, N., Daligault, J., Thomassé, S., Yeo, A.: A polynomial kernel for multicut in trees. In: STACS, pp. 183–194 (2009)Google Scholar
  4. 4.
    Chen, J., Fan, J., Kanj, I., Liu, Y., Zhang, F.: Multicut in trees viewed through the eyes of vertex cover. J. Comput. Syst. Sci. 78, 1637–1650 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chitnis, R., Hajiaghayi, M., Marx, D.: Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. In: SODA, pp. 1713–1725 (2012)Google Scholar
  6. 6.
    Chopra, S., Rao, M.: On the multiway cut polyhedron. Networks 21, 51–89 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Costa, M., Letocart, L., Roupin, F.: Minimal multicut and maximal integer multiflow: a survey. Eur. J. Oper. Res. 162, 55–69 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Costa, M.-C., Billionnet, A.: Multiway cut and integer flow problems in trees. Electron. Notes Discrete Math. 17, 105–109 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Deng, X., Lin, B., Zhang, C.: Multi-multiway cut problem on graphs of bounded branch width. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 315–324. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)CrossRefGoogle Scholar
  11. 11.
    Flüm, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2010)Google Scholar
  12. 12.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Guo, J., Niedermeier, R.: Fixed-parameter tractability and data reduction for multicut in trees. Networks 46, 124–135 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kanj, I.A., Lin, G., Liu, T., Tong, W., Xia, G., Xu, J., Yang, B., Zhang, F., Zhang, P., Zhu, B.: Algorithms for cut problems on trees. CoRR, abs/1304.3653 (2013)Google Scholar
  15. 15.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within \(2-\epsilon \). J. Comput. Syst. Sci. 74, 335–349 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Klein, P.N., Marx, D.: Solving Planar k -Terminal Cut in \(O(n^{c \sqrt{k}})\) time. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 569–580. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Levin, A., Segev, D.: Partial multicuts in trees. Theor. Comput. Sci. 369(1–3), 384–395 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Liu, H., Zhang, P.: On the generalized multiway cut in trees problem. J. Comb. Optim. 27(1), 65–77 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351, 394–406 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Marx, D.: A tight lower bound for planar multiway cut with fixed number of terminals. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 677–688. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. In: STOC, pp. 469–478 (2011)Google Scholar
  22. 22.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    West, D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Iyad Kanj
    • 1
  • Guohui Lin
    • 2
  • Tian Liu
    • 3
  • Weitian Tong
    • 2
  • Ge Xia
    • 4
  • Jinhui Xu
    • 5
  • Boting Yang
    • 6
  • Fenghui Zhang
    • 7
  • Peng Zhang
    • 8
  • Binhai Zhu
    • 9
  1. 1.School of ComputingDePaul UniversityChicagoUSA
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  3. 3.School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  4. 4.Department of Computing ScienceLafayette collegeEastonUSA
  5. 5.Department of Computer Science and EngineeringState University of New York at Buffalo (SUNY Buffalo)BuffaloUSA
  6. 6.Department of Computer ScienceUniversity of ReginaReginaCanada
  7. 7.Google KirklandKirklandUSA
  8. 8.School of Computer Science and TechnologyShandong UniversityJinanChina
  9. 9.Department of Computer ScienceMontana State UniversityBozemanUSA

Personalised recommendations