Genomes Containing Duplicates Are Hard to Compare

In this paper, we are interested in the algorithmic complexity of computing (dis)similarity measures between two genomes when they contain duplicated genes. In that case, there are usually two main ways to compute a given (dis)similarity measure

M

between two genomes

G

1

and

G

2

: the first model, that we will call the

matching model

, consists in computing a one-to-one correspondence between genes of

G

1

and genes of

G

2

, in such a way that

M

is optimized in the resulting permutation. The second model, called the

exemplar model

, consists in keeping in

G

1

(resp.

G

2

) exactly

one

copy of each gene, thus deleting all the other copies, in such a way that

M

is optimized in the resulting permutation. We present here different results concerning the algorithmic complexity of computing three different similarity measures (number of common intervals, MAD number and SAD number) in those two models, basically showing that the problem becomes

NP

-completeness for each of them as soon as genomes contain duplicates. In the case of MAD and SAD, we actually prove that, under both models, both MAD and SAD problems are APX-hard.