Abstract
In this paper, we study the problem Min-Max 2-Cluster Editing which asks for a modification of a given graph into two maximal cliques by inserting or deleting edges such that the maximum number k of the editing edges incident to any vertex is minimized. We show the NP-hardness of the problem and present a polynomial-time algorithm when k < n/4, in which n is number of vertices. In addition, we design a 2-approximation algorithm and a branching algorithm for finding an optimal solution. By experiments on random graphs, we show that the exact algorithm is much more efficient than a trivial one.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. J. ACM 55(5), 1–27 (2008)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial optimization problems and their approximability properties. Springer (1999)
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning, Special Issue on Clustering 56, 89–113 (2004)
Böcker, S., Damaschke, P.: Even faster parameterized cluster deletion and cluster editing. Information Processing Letters 111(14), 717–721 (2011)
Böcker, S., Briesemeister, S., Bui, Q.B.A., Truss, A.: Going weighted: Parameterized algorithm for cluster editing. Theor. Comput. Sci. 410(52), 5467–5480 (2009)
Bonizzoni, P., Della Vedova, G., Dondi, R.: A ptas for the minimum consensus clustering problem with a fixed number of clusters. In: Proc. Eleventh Italian Conference on Theoretical Computer Science (2009)
Bonizzoni, P., Della Vedova, G., Dondi, R., Jiang, T.: On the approximation of correlation clustering and consensus clustering. Journal of Computer and System Sciences 74(5), 671–696 (2008)
Chen, J., Meng, J.: A 2k Kernel for the Cluster Editing Problem. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 459–468. Springer, Heidelberg (2010)
Damaschke, P.: Bounded-Degree Techniques Accelerate Some Parameterized Graph Algorithms. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 98–109. Springer, Heidelberg (2009)
Damaschke, P.: Fixed-parameter enumerability of cluster editing and related problems. Theory Computing Syst. 46, 261–283 (2010)
Fellows, M.R., Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Graph-based data clustering with overlaps. Discrete Optimization 8(1), 2–17 (2011)
Filkov, V., Skiena, S.: Integrating microarray data by consensus clustering. International Journal on Artificial Intelligence Tools 13(4), 863–880 (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, NewYork (1979)
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.: Graph-modeled data clustering: Fixedparameter algorithms for clique generation. Theory Computing Syst. 38, 373–392 (2005)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39, 321–347 (2004)
Guo, J.: A more effective linear kernelization for cluster editing. Theor. Comput. Sci. 410, 718–726 (2009)
Harary, F.: On the notion of balance of a signed graph. Michigan Mathematical Journal 2(2), 143–146 (1953)
Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory of Computing Systems 47(1), 196–217 (2010)
Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: FOCS, pp. 17–27 (1980)
Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discr. Appl. Math. 144(1-2), 173–182 (2004)
Talmaciu, M., Nechita, E.: Recognition algorithm for diamond-free graphs. Informatica 18(3), 457–462 (2007)
Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, Cambridge (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, LH., Chang, MS., Wang, CC., Wu, B.Y. (2013). On the Min-Max 2-Cluster Editing Problem. In: Chang, RS., Jain, L., Peng, SL. (eds) Advances in Intelligent Systems and Applications - Volume 1. Smart Innovation, Systems and Technologies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35452-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-35452-6_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35451-9
Online ISBN: 978-3-642-35452-6
eBook Packages: EngineeringEngineering (R0)