An Improved Fixed-Parameter Algorithm for Minimum-Flip Consensus Trees

In computational phylogenetics, the problem of constructing a consensus tree for a given set of rooted input trees has frequently been addressed. In this article we study the <scp>Minimum-Flip Problem</scp>: the input trees are transformed into a binary matrix, and we want to find a perfect phylogeny for this matrix using a minimum number of flips, that is, corrections of single entries in the matrix. The graph-theoretical formulation of the problem is as follows: Given a bipartite graph <i>G</i> = (<i>Vt</i> ∪ <i>Vc</i>, <i>E</i>), the task is to find a minimum set of edge modifications such that the resulting graph has no induced path with four edges that starts and ends in <i>Vt</i>, where <i>Vt</i> corresponds to the taxa set and <i>Vc</i> corresponds to the character set. We present two fixed-parameter algorithms for the <scp>Minimum-Flip Problem</scp>, one with running time <i>O</i>(4.83<i>k</i> + <i>poly</i>(<i>m</i>, <i>n</i>)) and another one with running time <i>O</i>(4.42<i>k</i> + <i>poly</i>(<i>m</i>, <i>n</i>)) for <i>n</i> taxa, <i>m</i> characters, <i>k</i> flips, and <i>poly</i>(<i>m</i>, <i>n</i>) denotes a polynomial function in <i>m</i> and <i>n</i>. Additionally, we discuss several heuristic improvements. We also report computational results on phylogenetic data.