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 |).