Investigating the complexity of the double distance problems

Two genomes $$\mathbb {A}$$ and $$\mathbb {B}$$ over the same set of gene families form a canonical pair when each of them has exactly one gene from each family. Denote by $$n_*$$ the number of common families of $$\mathbb {A}$$ and $$\mathbb {B}$$ . 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 $$c_i$$ and $$p_j$$ be respectively the numbers of cycles of length i and of paths of length j in the breakpoint graph of genomes $$\mathbb {A}$$ and $$\mathbb {B}$$ . Then, the breakpoint distance of $$\mathbb {A}$$ and $$\mathbb {B}$$ is equal to $$n_*-\left( c_2+\frac{p_0}{2}\right)$$ . Similarly, when the considered rearrangements are those modeled by the double-cut-and-join (DCJ) operation, the rearrangement distance of $$\mathbb {A}$$ and $$\mathbb {B}$$ is $$n_*-\left( c+\frac{p_e }{2}\right)$$ , where c is the total number of cycles and $$p_e$$ 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 $$\sigma _k$$ distance, defined to be $$n_*-\left( c_2+c_4+\ldots +c_k+\frac{p_0+p_2+\ldots +p_{k-2}}{2}\right)$$ , and increasingly investigate the complexities of median and double distance for the $$\sigma _4$$ distance, then the  $$\sigma _6$$ 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 $$\sigma _4$$ distance, for solving the double distance under $$\sigma _4$$ and $$\sigma _6$$ distances we could devise linear time algorithms, which we present here.