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Theory of Computing Systems

, Volume 57, Issue 1, pp 1–35 | Cite as

On the Parameterized Complexity of Computing Balanced Partitions in Graphs

  • René van Bevern
  • Andreas Emil Feldmann
  • Manuel Sorge
  • Ondřej Suchý
Article

Abstract

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the Vertex Bisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n O(d). We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.

Keywords

Bisection NP-hard problems Problem kernelization Cliquewidth Treewidth reduction 

Notes

Acknowledgments

René van Bevern and Manuel Sorge gratefully acknowledge support by the DFG, research project DAPA, NI 369/12. Ondřej Suchý is also grateful for support by the DFG, research project AREG, NI 369/9. Part of Ondřej Suchý’s work was done while with TU Berlin.

The authors thank Bart M. P. Jansen, Stefan Kratsch, Rolf Niedermeier and the anonymous referees for helpful suggestions.

References

  1. 1.
    Andreev, K., Räcke, H.: Balanced graph partitioning. Theory Comput. Syst. 39 (6), 929–939 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arbenz, P.: Personal communication. ETH Zürich (2013)Google Scholar
  3. 3.
    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 International Workshop on Applied Parallel Computing (PARA 2006), volume 4699 of LNCS, pages 240–250. Springer (2007)Google 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)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H. L.: Kernelization: New upper and lower bound techniques. In: Proceedings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC 2009), volume 5917 of LNCS, pages 17–37. Springer (2009)Google Scholar
  6. 6.
    Bodlaender, H. L., Drange, P. G., Dregi, M. S., Fomin, F. V., Lokshtanov, D., Pilipczuk, M.: An O(c k n) 5-approximation algorithm for treewidth. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 499–508. IEEE Computer Society (2013)Google Scholar
  7. 7.
    Bodlaender, H. L., Jansen, B. M. P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math. 28 (1), 277–305 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brandes, U., Fleischer, D.: Vertex bisection is hard, too. J. Graph Algorithm. Appl. 13 (2), 119–131 (April 2009)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bui, T. N., Peck, A.: Partitioning planar graphs. SIAM J. Comput. 21 (2), 203–215 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bui, T. N., Chaudhuri, S., Leighton, F. T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7 (2), 171–191 (1987)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Chandran, L. S., Kavitha, T.: The treewidth and pathwidth of hypercubes. Discret. Math. 306 (3), 359–365 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen, J., Kanj, I. A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411 (40-42), 3736–3756 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. Appl. Math. 101 (1-3), 77–114 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Minimum bisection is fixed parameter tractable. In: Proceedings of the 46th Annual Symposium on the Theory of Computing (STOC 2014). To appear (2014)Google Scholar
  15. 15.
    Delling, D., Goldberg, A. V., Pajor, T., Werneck, R. F. F.: Customizable route planning. In: Proceedings of the 10th International Symposium on Experimental Algorithms (SEA 2011), volume 6630 of LNCS, pages 376–387. Springer (2011)Google Scholar
  16. 16.
    Delling, D., Goldberg, A. V., Razenshteyn, I., Werneck, R. F. F.: Exact combinatorial branch-and-bound for graph bisection. In: Proceedings of the 14th Workshop on Algorithms Engineering and Experiments (ALENEX 2012), pages 30–44 (2012)Google Scholar
  17. 17.
    Diestel, R., 4th edn: Graph Theory, Volume 173 of graduate texts in mathematics. Springer (2010)Google Scholar
  18. 18.
    Doucha, M., Kratochvíl, J.: Cluster vertex deletion: A parameterization between vertex cover and clique-width. In: Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS 2012), volume 7464 of LNCS, pages 348–359. Springer (2012)Google Scholar
  19. 19.
    Downey, R. G., Fellows, M. R.: Fundamentals of parameterized complexity. Springer (2013)Google Scholar
  20. 20.
    Enciso, R., Fellows, M. R., Guo, J., Kanj, I. A., Rosamond, F. A., Suchý, O.: What makes equitable connected partition easy. In: Proceedings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC 2009), volume 5917 of LNCS, pages 122–133. Springer (2009)Google Scholar
  21. 21.
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2001), volume 2204 of LNCS, pages 117–128. Springer (2001)Google Scholar
  22. 22.
    Feldmann, A. E.: Fast balanced partitioning is hard, even on grids and trees. Theor. Comput. Sci. 485, 61–68 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Feldmann, A. E., Foschini, L.: Balanced partitions of trees and applications. In: Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), volume 14 of LIPIcs, pages 100–111. Dagstuhl (2012)Google Scholar
  24. 24.
    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 2011), volume 6942 of LNCS, pages 143–154. Springer (2011)Google Scholar
  25. 25.
    Fellows, M. R., Hermelin, D., Rosamond, F. A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410 (1), 53–61 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)Google Scholar
  27. 27.
    Fomin, F. V., Golovach, P. A., Lokshtanov, D., Saurabh, S.: Algorithmic lower bounds for problems parameterized with clique-width. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pages 493–502, SIAM (2010)Google Scholar
  28. 28.
    Fomin, F. V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar \(\mathcal {F}\)-deletion: Approximation, kernelization and optimal FPT algorithms. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2012), pages 470–479. IEEE Computer Society (2012)Google Scholar
  29. 29.
    Fredman, M., Tarjan, R.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34 (3), 596–615 (1987)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Ganian, R., Obdržálek, J.: Expanding the expressive power of monadic second-order logic on restricted graph classes. In: Proceedings of the International Workshop on Combinatorial Algorithms (IWOCA 2013). Lect. Notes Comput. Sci. 8288, 164–177 (2013)Google Scholar
  31. 31.
    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
  32. 32.
    Garey, M. R., Johnson, D. S., Stockmeyer, L. J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1 (3), 237–267 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38 (1), 31–45 (2007)CrossRefGoogle Scholar
  34. 34.
    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
  35. 35.
    Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79 (1), 39–49 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Karypis, G., Kumar, V.: A parallel algorithm for multilevel graph partitioning and sparse matrix ordering. J. Parallel Distrib. Comput. 48 (1), 71–95 (1998)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Khot, S. A., Vishnoi, N. K.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into 1. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pages 53–62. IEEE Computer Society (2005)Google Scholar
  38. 38.
    Kloks, T: Treewidth – Computations and Approximations, volume 842 of LNCS. Springer (1994)Google Scholar
  39. 39.
    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 Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2002), volume 2573 of LNCS, pages 282–295. Springer (2002)Google Scholar
  40. 40.
    Kwatra, V., Schödl, A., Essa, I., Turk, G., Bobick, A.: Graphcut textures: Image and video synthesis using graph cuts. ACM Trans. Graph. 22 (3), 277–286 (2003)CrossRefGoogle Scholar
  41. 41.
    Lipton, R. J., Tarjan, R. E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    MacGregor, R. M.: On Partitioning a Graph: A theoretical and empirical study. PhD thesis. University of California, Berkeley (1978)Google Scholar
  43. 43.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351 (3), 394–406 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51 (1), 60–78 (2008)CrossRefGoogle Scholar
  45. 45.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithm. 9 (4), 30 (2013)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford University Press (2006)Google Scholar
  47. 47.
    Oum, S.: Approximating rank-width and clique-width quickly. ACM Trans. Algorithm. 5 (1) (2008)Google Scholar
  48. 48.
    Räcke, H. R.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC 2008), pages 255–264. ACM (2008)Google Scholar
  49. 49.
    Soumyanath, K., Deogun, J. S.: On the bisection width of partial k-trees. In: Proceedings of the 20th Southeastern Conference on Combinatorics, Graph Theory, and Computing, volume 74 of Congressus Numerantium, pages 25–37. Utilitas Mathematica Publishing (1990)Google Scholar
  50. 50.
    van Bevern, R., Feldmann, A. E., Sorge, M., Suchý, O.: On the parameterized complexity of computing graph bisections. In: Proceedings of the 39th International Workshop on Graph-Theoretic Concepts in Computer Science (WG ’13), volume 8165 of LNCS, pages 76–88. Springer (2013)Google Scholar
  51. 51.
    Werneck, R. F. F: Personal communication (2013). Microsoft Research Silicon ValleyGoogle Scholar
  52. 52.
    Wiegers, M.: The k-section of treewidth restricted graphs. In: Proceedings of the 15th International Symposium on Mathematical Foundations of Computer Science (MFCS 1990), volume 452 of LNCS, pages 530–537. Springer (1990)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • René van Bevern
    • 1
  • Andreas Emil Feldmann
    • 2
  • Manuel Sorge
    • 1
  • Ondřej Suchý
    • 3
  1. 1.Institut für Softwaretechnik und Theoretische InformatikBerlinGermany
  2. 2.Combinatorics, OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic

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