3 research outputs found
On the Complexity of the Median and Closest Permutation Problems
Genome rearrangements are events where large blocks of DNA exchange places
during evolution. The analysis of these events is a promising tool for
understanding evolutionary genomics, providing data for phylogenetic
reconstruction based on genome rearrangement measures. Many pairwise
rearrangement distances have been proposed, based on finding the minimum number
of rearrangement events to transform one genome into the other, using some
predefined operation. When more than two genomes are considered, we have the
more challenging problem of rearrangement-based phylogeny reconstruction. Given
a set of genomes and a distance notion, there are at least two natural ways to
define the "target" genome. On the one hand, finding a genome that minimizes
the sum of the distances from this to any other, called the median genome.
Finding a genome that minimizes the maximum distance to any other, called the
closest genome. Considering genomes as permutations, some distance metrics have
been extensively studied. We investigate median and closest problems on
permutations over the metrics: breakpoint, swap, block-interchange,
short-block-move, and transposition. In biological matters some values are
usually small, such as the solution value d or the number k of input
permutations. For each of these metrics and parameters d or k, we analyze the
closest and the median problems from the viewpoint of parameterized complexity.
We obtain the following results: NP-hardness for finding the median/closest
permutation for some metrics, even for k = 3; Polynomial kernels for the
problems of finding the median permutation of all studied metrics, considering
the target distance d as parameter; NP-hardness result for finding the closest
permutation by short-block-moves; FPT algorithms and infeasibility of
polynomial kernels for finding the closest permutation for some metrics
parameterized by the target distance d