On the Matrix Median Problem

The Genome Median Problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. Recently, Feijao and Meidanis extended the algebraic theory for genome rearrangement to allow for linear chromosomes, thus yielding a new rearrangement model (the algebraic model), very close to the celebrated DCJ model. In this paper, we study the genome median problem under the algebraic model, whose complexity is currently open, proposing a more general form of the problem, the matrix median problem. It is known that, for any metric distance, at least one of the corners is a 4/3-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. From the application point of view, it is usually more interesting to locate medians farther from the corners. We also show a fourth median candidate that gives better results in cases we tried. However, we do not have proven bounds for this fourth candidate yet.