The Complexity of Tree Partitioning

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

Given a tree T on n vertices, and \(k, b, s_1, \ldots , s_b \in \mathbb {N}\), the Tree Partitioning problem asks if at most k edges can be removed from T so that the resulting components can be grouped into b groups such that the number of vertices in group i is \(s_i\), for \(i =1, \ldots , b\). The case when \(s_1=\cdots =s_b =n/b\), referred to as the Balanced Tree Partitioning problem, was shown to be \(\mathcal {NP}\)-complete for trees of maximum degree at most 5, and the complexity of the problem for trees of maximum degree 4 and 3 was posed as an open question. The parameterized complexity of Balanced Tree Partitioning was also posed as an open question in another work.

In this paper, we answer both open questions negatively. We show that Balanced Tree Partitioning (and hence, Tree Partitioning) is \(\mathcal {NP}\)-complete for trees of maximum degree 3, thus closing the door on the complexity of Balanced Tree Partitioning, as the simple case when T is a path is in \(\mathcal P\). In terms of the parameterized complexity of the problems, we show that both Balanced Tree Partitioning and Tree Partitioning are W[1]-complete. Finally, using a compact representation of the solution space for an instance of the problem, we present a dynamic programming algorithm for Tree Partitioning (and hence, for Balanced Tree Partitioning) that runs in subexponential-time \(2^{O(\sqrt{n})}\), adding a natural problem to the list of problems that can be solved in subexponential time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    An, Z., Feng, Q., Kanj, I., Xia, G.: The Complexity of Tree Partitioning. http://arxiv.org/abs/1704.05896
  2. 2.
    Andreev, K., Räcke, H.: Balanced graph partitioning. Theory of Computing Systems 39(6), 929–939 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arbenz, P., van Lenthe, G., Mennel, U., Müler, R., Sala, M.: Multi-level \(\mu \)-finite element analysis for human bone structures. In: PARA 2006, pp. 240–250 (2006)Google Scholar
  4. 4.
    Bhatt, S., Leighton, F.: A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences 28(2), 300–343 (1984)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boscznay, Á.: On the lower estimation of non-averaging sets. Acta Mathematica Hungariga 53(1-1), 155–157 (1989)Google Scholar
  6. 6.
    Chen, J., Kanj, I., Perkovic, L., Sedgwick, E., Xia, G.: Genus characterizes the complexity of certain graph problems: Some tight results. Journal of Computer and System Sciences 73(6), 892–907 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, Y., Flum, J., Grohe, M.: Machine-based methods in parameterized complexity theory. Theoretical Computer Science 339(2–3), 167–199 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Delling, D., Goldberg, A.V., Pajor, T., Werneck, R.F.: Customizable route planning. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 376–387. Springer, Heidelberg (2011). doi:10.1007/978-3-642-20662-7_32 CrossRefGoogle Scholar
  9. 9.
    Demaine, E., Fomin, F., Hajiaghayi, M., Thilikos, D.: Subexponential parameterized algorithms on bounded-genus graphs and \(H\)-minor-free graphs. J. ACM 52, 866–893 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diestel, R.: Graph Theory, 4th edn. Springer (2012)Google Scholar
  11. 11.
    Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, New York (2013)CrossRefMATHGoogle Scholar
  12. 12.
    Feldmann, A.: Balanced partitions of grids and related graphs. Ph.D. thesis, ETH, Zurich, Switzerland (2012)Google Scholar
  13. 13.
    Feldmann, A., Foschini, L.: Balanced partitions of trees and applications. Algorithmica 71(2), 354–376 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Feldmann, A., Widmayer, P.: An \(O(n^4)\) time algorithm to compute the bisection width of solid grid graphs. Algorithmica 71(1), 181–200 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fomin, F., Kolay, S., Lokshtanov, D., Panolan, F., Saurabh, S.: Subexponential algorithms for rectilinear steiner tree and arborescence problems. In: SoCG 2016, pp. 39: 1–39: 15 (2016)Google Scholar
  16. 16.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)MATHGoogle Scholar
  17. 17.
    Hardy, G., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proceedings of the London Mathematical Society 17(2), 75–115 (1918)CrossRefMATHGoogle Scholar
  18. 18.
    Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences 79(1), 39–49 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Klein, P., Marx, D.: A subexponential parameterized algorithm for subset TSP on planar graphs. SODA 2014, 1812–1830 (2014)MathSciNetGoogle Scholar
  20. 20.
    MacGregor, R.: On partitioning a graph: a theoretical and empirical study. Ph.D. thesis, University of California at Berkeley, California, USA (1978)Google Scholar
  21. 21.
    Räcke, H., Stotz, R.: Improved approximation algorithms for balanced partitioning problems. In STACS 2016, pp. 58: 1–58: 14 (2016)Google Scholar
  22. 22.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar
  23. 23.
    van Bevern, R., Feldmann, A., Sorge, M., Suchý, O.: On the parameterized complexity of computing balanced partitions in graphs. Theory of Computing Systems 57(1), 1–35 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Information Science and EngineeringCentral South UniversityChangsha ShiChina
  2. 2.School of ComputingDePaul UniversityChicagoUSA
  3. 3.Dept. of Computer ScienceLafayette CollegeEastonUSA

Personalised recommendations