A Cubic-Vertex Kernel for Flip Consensus Tree
Résumé
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 I -graph) pound. 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) a <...|V (t) |a <...|V (c) |). Subsequently, Bocker 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) |a <...|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) a <...|V (c) |(2)|V (t) |).