Fast Balanced Partitioning Is Hard Even on Grids and Trees

  • Andreas Emil Feldmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfactory approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is unavoidable. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a general reduction which identifies some sufficient conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite two-dimensional grid without holes. We apply the reduction to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfactory ratio. We also consider solutions in which the sets may deviate from being equal-sized. Our reduction is applied to grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase when the limit on the set sizes decreases. These are the first bicriteria inapproximability results for the k-BALANCED PARTITIONING problem.


Planar Graph Polynomial Time Algorithm Graph Class Grid Graph Majority Colour 
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.


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  1. 1.
    Andreev, K., Räcke, H.: Balanced graph partitioning. Theory of Computing Systems 39(6), 929–939 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arbenz, P., van Lenthe, G., Mennel, U., Müller, R., Sala, M.: Multi-level μ-finite element analysis for human bone structures. In: Proceedings of the 8th Workshop on State-of-the-art in Scientific and Parallel Computing (PARA), pp. 240–250 (2007)Google Scholar
  3. 3.
    Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC), pp. 222–231 (2004)Google Scholar
  4. 4.
    Bhatt, S., Leighton, F.T.: A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences 28(2), 300–343 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chevalier, C., Pellegrini, F.: PT-Scotch: A tool for efficient parallel graph ordering. Parallel Computing 34(68), 318–331 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Delling, D., Goldberg, A., Pajor, T., Werneck, R.: Customizable route planning. Experimental Algorithms, 376–387 (2011)Google Scholar
  7. 7.
    Díaz, J., Serna, M.J., Torán, J.: Parallel approximation schemes for problems on planar graphs. Acta Informatica 33(4), 387–408 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Diestel, R., Jensen, T.R., Gorbunov, K.Y., Thomassen, C.: Highly connected sets and the excluded grid theorem. Journal of Combinatorial Theory, Series B 75(1), 61–73 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Diks, K., Djidjev, H.N., Sykora, O., Vrto, I.: Edge separators of planar and outerplanar graphs with applications. Journal of Algorithms 14(2), 258–279 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, USA (2005)zbMATHGoogle Scholar
  11. 11.
    Even, G., Naor, J., Rao, S., Schieber, B.: Fast approximate graph partitioning algorithms. SIAM Journal on Computing 28(6), 2187–2214 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Feldmann, A.E.: Balanced Partitioning of Grids and Related Graphs: A Theoretical Study of Data Distribution in Parallel Finite Element Model Simulations. PhD thesis, ETH Zurich, Diss.-Nr. ETH: 20371 (April 2012)Google Scholar
  13. 13.
    Feldmann, A.E., Das, S., Widmayer, P.: Restricted cuts for bisections in solid grids: A proof via polygons. In: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 143–154 (2011)Google Scholar
  14. 14.
    Feldmann, A.E., Foschini, L.: Balanced partitions of trees and applications. In: 29th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 100–111 (2012)Google Scholar
  15. 15.
    Feldmann, A.E., Widmayer, P.: An O(n 4) time algorithm to compute the bisection width of solid grid graphs. In: Proceedings of the 19th Annual European Symposium on Algorithms (ESA), pp. 143–154 (2011)Google Scholar
  16. 16.
    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
  17. 17.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoretical Computer Science 1(3), 237–267 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Karypis, G., Kumar, V.: METIS-unstructured graph partitioning and sparse matrix ordering system, version 2.0. Technical report, University of Minnesota (1995)Google Scholar
  19. 19.
    Khot, S.A., Vishnoi, N.K.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 53–62 (2005)Google Scholar
  20. 20.
    Klein, P., Plotkin, S., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pp. 682–690 (1993)Google Scholar
  21. 21.
    Krauthgamer, R., Naor, J., Schwartz, R.: Partitioning graphs into balanced components. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 942–949 (2009)Google Scholar
  22. 22.
    Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lipton, R., Tarjan, R.: Applications of a planar separator theorem. SIAM Journal on Computing 9, 615–627 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    MacGregor, R.M.: On Partitioning a Graph: a Theoretical and Empirical Study. PhD thesis, University of California, Berkeley (1978)Google Scholar
  25. 25.
    Papadimitriou, C., Sideri, M.: The bisection width of grid graphs. Theory of Computing Systems 29, 97–110 (1996)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Park, J.K., Phillips, C.A.: Finding minimum-quotient cuts in planar graphs. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pp. 766–775 (1993)Google Scholar
  27. 27.
    Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC) (2008)Google Scholar
  28. 28.
    Simon, H.D., Teng, S.H.: How good is recursive bisection? SIAM Journal on Scientific Computing 18(5), 1436–1445 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Emil Feldmann
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichSwitzerland

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