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Applying Modular Decomposition to Parameterized Cluster Editing Problems

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Abstract

A graph G is said to be a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs), and a cluster graph if G is a disjoint union of cliques (complete subgraphs). In this work, we study the parameterized versions of the NP-hard Bicluster Graph Editing and Cluster Graph Editing problems. The former consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed (Bicluster(k) Graph Editing problem), this problem is FPT, and can be solved in O(4k nm) time by a standard search tree algorithm. We develop an algorithm of time complexity O(4k+n+m), which uses a strategy based on modular decomposition techniques; we slightly generalize the original problem as the input graph is not necessarily bipartite. The algorithm first builds a problem kernel with O(k 2) vertices in O(n+m) time, and then applies a bounded search tree. We also show how this strategy based on modular decomposition leads to a new way of obtaining a problem kernel with O(k 2) vertices for the Cluster(k) Graph Editing problem, in O(n+m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm of time O(1.92k+n 3) for this problem was presented by Gramm et al. (Theory Comput. Syst. 38(4), 373–392, 2005, Algorithmica 39(4), 321–347, 2004). In their solution, a problem kernel with O(k 2) vertices is built in O(n 3) time.

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References

  1. Amit, N.: The bicluster graph editing problem. M.Sc. Thesis, Tel Aviv University (2004)

  2. Bauer, H., Möhring, R.H.: A fast algorithm for the decomposition of graphs and posets. Math. Oper. Res. 8, 170–184 (1983)

    Article  MathSciNet  Google Scholar 

  3. Bretscher, A., Corneil, D., Habib, M., Paul, C.: A simple linear time lexBFS cograph recognition algorithm. In: WG 2003. Lecture Notes in Computer Science, vol. 2880, pp. 119–130 (2003)

  4. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58, 171–176 (1996)

    Article  MATH  Google Scholar 

  5. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dahlhaus, E., Gustedt, J., McConnell, R.M.: Efficient and practical algorithms for sequential modular decomposition. J. Algorithms 41, 360–387 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dantas da Silva, M., Protti, F., Szwarcfiter, J.L.: Applying modular decomposition to parameterized bicluster editing. In: IWPEC 2006—Second International Workshop on Parameterized and Exact Computation, Zurich, Switzerland. Lecture Notes in Computer Science, vol. 4169, pp. 1–12 (2006)

  8. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput. 24(4), 873–921 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: on completeness for W[1]. Theor. Comput. Sci. 141, 109–131 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Google Scholar 

  11. Fellows, M., Langston, M., Rosamond, F., Shaw, P.: Efficient parameterized preprocessing for cluster editing. Manuscript (2006)

  12. Fernau, H.: Parameterized Algorithms: A Graph-Theoretic Approach. University of Newcastle (2005)

  13. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)

    Google Scholar 

  14. Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hung. 18, 26–66 (1967)

    Article  MathSciNet  Google Scholar 

  15. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: fixed-parameter algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo, J.: A more effective linear kernelization for Cluster Editing. In: Proceedings of the 1st International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies (ESCAPE 2007), Hangzhou, China, April 2007. Lecture Notes in Computer Science (2007)

  18. Habib, M., Montgolfier, F., Paul, C.: A simple linear-time modular decomposition algorithm for graphs, using order extension. In: 9th Scandinavian Workshop on Algorithm Theory (SWAT 2004). Lecture Notes in Computer Science, vol. 3111, pp. 187–198 (2004)

  19. Möhring, R.H., Radermacher, F.J.: Substitution decomposition and connections with combinatorial optimization. Ann. Discret. Math. 19, 257–356 (1984)

    Google Scholar 

  20. McConnell, R.M., Spinrad, J.P.: Linear-time modular decomposition and efficient transitive orientation of comparability graphs. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 5, pp. 536–545 (1994)

  21. McConnell, R.M., Spinrad, J.P.: Ordered vertex partitioning. Discret. Math. Theor. Comput. Sci. 4, 45–60 (2000)

    MATH  MathSciNet  Google Scholar 

  22. Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discret. Appl. Math. 113, 109–128 (1999)

    Article  MathSciNet  Google Scholar 

  23. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  24. Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parameter-tractable algorithms. Inf. Process. Lett. 73, 125–129 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discret. Appl. Math. 144, 173–182 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática and Núcleo de Computação Eletrônica, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brazil

    Fábio Protti & Jayme Luiz Szwarcfiter

  2. COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21945-970, Rio de Janeiro, RJ, Brazil

    Maise Dantas da Silva & Jayme Luiz Szwarcfiter

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  1. Fábio Protti
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  2. Maise Dantas da Silva
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  3. Jayme Luiz Szwarcfiter
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Correspondence to Fábio Protti.

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Protti, F., Dantas da Silva, M. & Szwarcfiter, J.L. Applying Modular Decomposition to Parameterized Cluster Editing Problems. Theory Comput Syst 44, 91–104 (2009). https://doi.org/10.1007/s00224-007-9032-7

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  • DOI: https://doi.org/10.1007/s00224-007-9032-7

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