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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
Article
  • 185 Downloads

Abstract

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.

Keywords

Parameterized algorithm Exact algorithm Cluster graph Graph modification 

Notes

Acknowledgments

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.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.National Chung Cheng UniversityChiayiTaiwan, ROC

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