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

A Cubic-Vertex Kernel for Flip Consensus Tree



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 |).


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



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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