Abstract
The Labeled Correlation Clustering problem, a variant of Correlation Clustering problem, is defined and studied in this paper. It is shown that the problem is NP-complete, and an approximation algorithm is given. For the case when a parameter is fixed, a better approximation algorithm is proposed, and, for a simple fragment of that problem, a PTime algorithm is introduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
By Papadimitriou and Yannakakis (1988), given optimization problems \(A\) and \(B\), and their cost functions \(c_A\) and \(c_B\), a pair of functions \(f\) and \(g\) is an L-reduction from \(A\) to \(B\), if (i) \(f\) and \(g\) are computable in PTime, (ii) for an instance \(x\) of \(A\), then \(f(x)\) is an instance of B, (iii) for a solution \(s\) of \(f(x), g(s)\) is a solution of \(x\), (iv) there exists two positive constants \(\alpha \) and \(\beta \) such that \(c_B(\mathsf{{opt} }_{f(x)})\le \alpha \cdot c_A(\mathsf{{opt} }_{x})\) and \(|c_A(\mathsf{{opt} }_{x})-c_A(g(s))|\le \beta \cdot |c_B(\mathsf{{opt} }_{f(x)})-c_B(s)|\), where \(\mathsf{{opt} }_{-}\) are optimal solutions.
References
Ailon N, Charikar M, Newman A (2008) Aggregating inconsistent information: ranking and clustering. J ACM 55(2):23:1–23:27
Bansal N, Blum A, Chawla S (2004) Correlation clustering. Mach Learn 56(1–3):89–113
Broersma H, Li X, Woeginger G, Zhang S (2005) Paths and cycles in colored graphs. Australas J Comb 31:299–311
BrüGgemann T, Monnot J, Woeginger GJ (2003) Local search for the minimum label spanning tree problem with bounded color classes. Oper Res Lett 31(3):195–201
Carr RD, Doddi S, Konjevod G, Marathe M (2000) On the red-blue set cover problem. In: Proceedings of the 11th ACM-SIAM SODA pp 345–353
Charikar M, Guruswami V, Wirth A (2005) Clustering with qualitative information. J Comput Syst Sci 71(3):360–383
Demaine ED, Emanuel D, Fiat A, Immorlica N (2006) Correlation clustering in general weighted graphs. Theor Comput Sci 361(2):172–187
Fellows MR, Guo J, Kanj I (2010) The parameterized complexity of some minimum label problems. J Comput Syst Sci 76:727–740
Garey MR, Johnson DS (1990) Computers intractability: a guide to the theory of NP-completeness. W. H. Freeman, New York
Hassin R, Monnot J, Segev D (2007) Approximation algorithms and hardness results for labeled connectivity problems. J Comb Optim 14:437–453
Jha S, Sheyner O, Wing J (2002) Two formal analys s of attack graphs. In: Proceedings of the 15th IEEE workshop on computer security foundations
Malioutov I, Barzilay R (2006) Minimum cut model for spoken lecture segmentation. In: Proceedings of COLING-ACL
Papadimitriou C, Yannakakis M (1988) Optimization, approximation, and complexity classes. In: Proceedings of the 20th ACM symposium on theory of computing pp 229–234
Soon WM, Ng HT, Lim DCY (2001) A machine learning approach to coreference resolution of noun phrases. Comput Linguist 27(4):521–544
Zhang P, Cai JY, Tang LQ, Zhao WB (2011) Approximation and hardness results for label cut and related problems. J Comb Optim 21:192–208
Acknowledgments
The work in this paper was partially supported by the National Basic Research (973) Program of China under Grant No. 2012CB316202, the National Natural Science Foundation of China under Grant No. 61003046 and No. 6111113089. We would like to thank anonymous reviewers for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, X., Li, J. Algorithms and complexity results for labeled correlation clustering problem. J Comb Optim 29, 488–501 (2015). https://doi.org/10.1007/s10878-013-9607-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-013-9607-y