Algorithmica

, Volume 68, Issue 1, pp 81–108 | Cite as

A Cubic-Vertex Kernel for Flip Consensus Tree

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Abstract

Given a bipartite graph G=(V c ,V t ,E) and a nonnegative integer k, the NP-complete Minimum-Flip Consensus Tree problem asks whether G can be transformed, using up to k edge insertions and deletions, into a graph that does not contain an induced P 5 with its first vertex in V t (a so-called M-graph or Σ-graph). This problem plays an important role in computational phylogenetics, V c standing for the characters and V t standing for taxa. Chen et al. (IEEE/ACM Trans. Comput. Biol. Bioinform. 3:165–173, 2006). showed that Minimum-Flip Consensus Tree is NP-complete and presented a parameterized algorithm with running time O(6 k ⋅|V t |⋅|V c |). Subsequently, Böcker et al. (ACM Trans. Algorithms 8:7:1–7:17, 2012) presented a refined search tree algorithm with running time O(4.42 k (|V t |+|V c |)+|V t |⋅|V c |). We continue the study of Minimum-Flip Consensus Tree parameterized by k. Our main contribution are polynomial-time executable data reduction rules yielding a problem kernel with O(k 3) vertices. In addition, we present an improved search tree algorithm with running time O(3.68 k ⋅|V c |2|V t |).

Keywords

Computational phylogenetics Perfect phylogeny NP-hard problem Edge-modification problem Fixed-parameter tractability Polynomial-time preprocessing 

Notes

Acknowledgements

We are grateful to Rolf Niedermeier, Jiong Guo, and to the anonymous reviewers of FSTTCS ’08 and Algorithmica for discussions and comments improving the presentation of this work.

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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