Investigating the complexity of the double distance problems

Two genomes [Formula: see text] and [Formula: see text] over the same set of gene families form a canonical pair when each of them has exactly one gene from each family. Denote by [Formula: see text] the number of common families of [Formula: see text] and [Formula: see text]. Different distances of canonical genomes can be derived from a structure called breakpoint graph, which represents the relation between the two given genomes as a collection of cycles of even length and paths. Let [Formula: see text] and [Formula: see text] be respectively the numbers of cycles of length i and of paths of length j in the breakpoint graph of genomes [Formula: see text] and [Formula: see text]. Then, the breakpoint distance of [Formula: see text] and [Formula: see text] is equal to [Formula: see text]. Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance of [Formula: see text] and [Formula: see text] is [Formula: see text], where c is the total number of cycles and [Formula: see text] is the total number of paths of even length.

The distance formulation is a basic unit for several other combinatorial problems related to genome evolution and ancestral reconstruction, such as median or double distance. Interestingly, both median and double distance problems can be solved in polynomial time for the breakpoint distance, while they are NP-hard for the rearrangement distance. One way of exploring the complexity space between these two extremes is to consider a [Formula: see text] distance, defined to be [Formula: see text], and increasingly investigate the complexities of median and double distance for the [Formula: see text] distance, then the [Formula: see text] distance, and so on.

While for the median much effort was done in our and in other research groups but no progress was obtained even for the [Formula: see text] distance, for solving the double distance under [Formula: see text] and [Formula: see text] distances we could devise linear time algorithms, which we present here.