Journal of Combinatorial Optimization

, Volume 33, Issue 2, pp 373–388 | Cite as

An improved parameterized algorithm for the p-cluster vertex deletion problem

  • Bang Ye WuEmail author
  • Li-Hsuan Chen


In the p-Cluster Vertex Deletion problem, we are given a graph \(G=(V,E)\) and two parameters k and p, and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let \(r=p/k\). In this paper, we design a branching algorithm with time complexity \(O(\alpha ^k+|V||E|)\), where \(\alpha \) depends on r and has a rough upper bound \(\min \{1.618^{1+r},2\}\). With a more precise analysis, we show that \(\alpha =1.28\cdot 3.57^{r}\) for \(r\le 0.219\); \(\alpha =(1-r)^{r-1}r^{-r}\) for \(0.219< r<1/2\); and \(\alpha =2\) for \(r\ge 1/2\), respectively. Our algorithm also works with the same time complexity for the variant that the number of clusters is at most p. Our result improves the previous best time complexity \(O^*(1.84^{p+k})\) and implies that for fixed p the problem can be solved as efficiently as Vertex Cover.


Parameterized algorithm Exact algorithm Cluster graph Graph modification 



This work was supported in part by NSC 101-2221-E-194-025-MY3 and MOST 103-2221-E-194-025-MY3 from Ministry of Science and Technology, Taiwan, ROC.


  1. Abu-Khzam FN (2010) A kernelization algorithm for \(d\)-hitting set. J Comput Syst Sci 76(7):524–531MathSciNetCrossRefzbMATHGoogle Scholar
  2. Böcker S, Briesemeister S, Bui Q, Truss A (2009) Going weighted: Parameterized algorithms for cluster editing. Theor Comput Sci 410(52):5467–5480MathSciNetCrossRefzbMATHGoogle Scholar
  3. Böcker S, Damaschke P (2011) Even faster parameterized cluster deletion and cluster editing. Inf Process Lett 111(14):717–721MathSciNetCrossRefzbMATHGoogle Scholar
  4. Boral A, Cygan M, Kociumaka T, Pilipczuk M (2015) A fast branching algorithm for cluster vertex deletion. Theory Comput Syst. doi: 10.1007/s00224-015-9631-7
  5. Chen J, Kanj IA, Xia G (2010) Improved upper bounds for vertex cover. Theor Comput Sci 411(40–42):3736–3756MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chen J, Meng J (2012) A \(2k\) kernel for the cluster editing problem. J Comput Syst Sci 78(1):211–220MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chen LH, Chang MS, Wang CC, Wu BY (2013) On the min-max 2-cluster editing problem. J Inf Sci Eng 29:1109–1120MathSciNetGoogle Scholar
  8. Damaschke P (2009) Bounded-degree techniques accelerate some parameterized graph algorithms. In: Chen J, Fomin F (eds) Parameterized and exact computation, lecture notes in computer science, vol 5917. Springer, Berlin Heidelberg, pp 98–109CrossRefGoogle Scholar
  9. Damaschke P (2010) Fixed-parameter enumerability of cluster editing and related problems. Theory Comput Syst 46:261–283MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fellows MR, Guo J, Komusiewicz C, Niedermeier R, Uhlmann J (2011) Graph-based data clustering with overlaps. Discret Optim 8(1):2–17MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fomin FV, Gaspers S, Kratsch D, Liedloff M, Saurabh S (2010) Iterative compression and exact algorithms. Theor Comput Sci 411(7–9):1045–1053MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fomin FV, Grandoni F, Kratsch D (2009) A measure & conquer approach for the analysis of exact algorithms. J ACM 56(5):25:1–25:32MathSciNetCrossRefzbMATHGoogle Scholar
  13. Fomin FV, Kratsch S, Pilipczuk M, Pilipczuk M, Villanger Y (2014) Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J Comput Syst Sci 80(7):1430–1447MathSciNetCrossRefzbMATHGoogle Scholar
  14. Gramm J, Guo J, Hüffner F, Niedermeier R (2005) Graph-modeled data clustering: exact algorithms for clique generation. Theory Comput Syst 38(4):373–392MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gramm J, Guo J, Hüffner F, Niedermeier R (2004) Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39:321–347MathSciNetCrossRefzbMATHGoogle Scholar
  16. Guo J (2009) A more effective linear kernelization for cluster editing. Theor Comput Sci 410(8–10):718–726MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hüffner F, Komusiewicz C, Moser H, Niedermeier R (2010) Fixed-parameter algorithms for cluster vertex deletion. Theory Comput Syst 47:196–217MathSciNetCrossRefzbMATHGoogle Scholar
  18. Komusiewicz C, Uhlmann J (2012) Cluster editing with locally bounded modifications. Discret Appl Math 160(15):2259–2270MathSciNetCrossRefzbMATHGoogle Scholar
  19. Niedermeier R, Rossmanith P (2000) A general method to speed up fixed-parameter-tractable algorithms. Inf Process Lett 73(3–4):125–129MathSciNetCrossRefzbMATHGoogle Scholar
  20. Shamir R, Sharan R, Tsur D (2004) Cluster graph modification problems. Discret Appl Math 144(1–2):173–182MathSciNetCrossRefzbMATHGoogle Scholar
  21. Wu BY, Chen LH (2015) Parameterized algorithms for the 2-clustering problem with minimum sum and minimum sum of squares objective functions. Algorithmica 72:818–835MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.National Chung Cheng UniversityChiayiTaiwan, ROC

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