Abstract
We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.
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Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Proc. 10th WADS. Lecture Notes in Computer Science, vol. 4619, pp. 434–445. Springer, Berlin (2007)
Abu-Khzam, F.N., Fernau, H.: Kernels: Annotated, proper and induced. In: Proc. 2nd IWPEC. Lecture Notes in Computer Science, vol. 4169, pp. 264–275. Springer, Berlin (2006)
Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. In: Proc. 37th STOC, pp. 684–693. Assoc. Comput. Math., New York (2005)
Ailon, N., Charikar, M., Newman, A.: Proofs of conjectures in “Aggregating inconsistent information: Ranking and clustering”. Technical Report TR-719-05, Department of Computer Science, Princeton University (2005)
Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: A fixed-parameter approach for weighted cluster editing. In: Proc. 6th APBC. Series on Advances in Bioinformatics and Computational Biology, vol. 5, pp. 211–220. Imperial College Press, London (2008)
Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: Parameterized algorithms for cluster editing. In: Proc. 2nd COCOA. Lecture Notes in Computer Science, vol. 5165, pp. 1–12. Springer, Berlin (2008)
Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)
Cai, M.-C., Deng, X., Zang, W.: An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput. 30(6), 1993–2007 (2001)
Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Proc. 31st MFCS. Lecture Notes in Computer Science, vol. 4162, pp. 238–249. Springer, Berlin (2006)
Dehne, F., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: Implementations and experiments. In: Proc. 2nd IWPEC. Lecture Notes in Computer Science, vol. 4169, pp. 13–24. Springer, Berlin (2006)
Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. In: Proc. 6th CIAC. Lecture Notes in Computer Science, vol. 3998, pp. 320–331. Springer, Berlin (2006)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Fellows, M.R., Langston, M.A., Rosamond, F.A., Shaw, P.: Efficient parameterized preprocessing for cluster editing. In: Proc. 16th FCT. Lecture Notes in Computer Science, vol. 4639, pp. 312–321. Springer, Berlin (2007)
Fernau, H.: Parameterized algorithms for hitting set: The weighted case. In: Proc. 6th CIAC. Lecture Notes in Computer Science, vol. 3998, pp. 332–343. Springer, Berlin (2006)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)
Fomin, F.V., Gaspers, S., Kratsch, D., Liedloff, M., Saurabh, S.: Iterative compression and exact algorithms. In: Proc. 33rd MFCS. Lecture Notes in Computer Science, vol. 5162, pp. 335–346. Springer, Berlin (2008)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18(5), 1013–1036 (1989)
Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. Theory Comput. 2, 249–266 (2006)
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)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Exact algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)
Guo, J.: A more effective linear kernelization for cluster editing. In: Proc. 1st ESCAPE. Lecture Notes in Computer Science, vol. 4614, pp. 36–47. Springer, Berlin (2007). Long version to appear in Theoretical Computer Science
Guo, J., Moser, H., Niedermeier, R.: Iterative compression for exactly solving NP-hard minimization problems. In: Algorithmics of Large and Complex Networks. Lecture Notes in Computer Science. Springer, Berlin (2008, to appear)
Hüffner, F.: Algorithms and experiments for parameterized approaches to hard graph problems. Ph.D. thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena (2007)
Hüffner, F., Niedermeier, R., Wernicke, S.: Techniques for practical fixed-parameter algorithms. Comput. J. 51(1), 7–25 (2008)
Jansen, K., Scheffler, P., Woeginger, G.: The disjoint cliques problem. RAIRO Rech. Opér. 31(1), 45–66 (1997)
Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)
Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: Proc. 20th ICALP. Lecture Notes in Computer Science, vol. 700, pp. 40–51. Springer, Berlin (1993)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications, vol. 31. Oxford University Press, London (2006)
Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 320–331 (2003)
Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parameter-tractable algorithms. Inf. Process. Lett. 73, 125–129 (2006)
Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. (2008, to appear)
Rahmann, S., Wittkop, T., Baumbach, J., Martin, M., Truß, A., Böcker, S.: Exact and heuristic algorithms for weighted cluster editing. In: Proc. 6th CSB. Computational Systems Bioinformatics, vol. 6, pp. 391–401. Imperial College Press, London (2007)
Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)
Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)
Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)
Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Department of Computer and Information Science, Linköpings universitet (2007)
van Zuylen, A., Williamson, D.P.: Deterministic algorithms for rank aggregation and other ranking and clustering problems. In: Proc. 5th WAOA. Lecture Notes in Computer Science, vol. 4927, pp. 260–273. Springer, Berlin (2008)
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An extended abstract of this paper appeared in Proceedings of the 8th Latin American Theoretical Informatics Symposium (LATIN 2008), April 7–11, 2008, Armação dos Búzios, Brazil, volume 4957 in Lecture Notes in Computer Science, pp. 711–722, Springer, 2008.
The research of F. Hüffner was supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4, and the Edmond J. Safra foundation.
The research of C. Komusiewicz was supported by a Ph.D. fellowship of the Carl-Zeiss-Stiftung.
The research of H. Moser was supported by the Deutsche Forschungsgemeinschaft, project ITKO (iterative compression for solving hard network problems), NI 369/5, and project AREG (algorithms for generating quasi-regular structures in graphs), NI 369/9.
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Hüffner, F., Komusiewicz, C., Moser, H. et al. Fixed-Parameter Algorithms for Cluster Vertex Deletion. Theory Comput Syst 47, 196–217 (2010). https://doi.org/10.1007/s00224-008-9150-x
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DOI: https://doi.org/10.1007/s00224-008-9150-x