Compromise or optimize? The breakpoint anti-median

The median of k≥3 genomes was originally defined to find a compromise genome indicative of a common ancestor. However, in gene order comparisons, the usual definitions based on minimizing the sum of distances to the input genomes lead to degenerate medians reflecting only one of the input genomes. "Near-medians", consisting of equal samples of gene adjacencies from all the input genomes, were designed to restore the idea of compromise to the median problem.

We explore adjacency sampling constructions in full generality in the case k=3, with given overlapping sets of adjacencies in the three genomes, where all adjacencies in two-way or three-way overlaps are included in the sample. We require the construction to be maximal, in the sense that no additional proportion of adjacencies from any of the genomes may be added without violating the local linearity of the genome. We discover that in incorporating as many adjacencies as possible, evenly from all the input genomes, we are actually maximizing, rather than minimizing, the sum of distances over all other maximal sampling schemes.

We propose to explore compromise instead of parsimony as the organizing principle for the small phylogeny problem.