Inapproximability of (1,2)-exemplar distance.

Given two genomes possibly with duplicate genes, the exemplar distance problem is that of removing all but one copy of each gene in each genome, so as to minimize the distance between the two reduced genomes according to some measure. Let $((s,t))$-exemplar distance denote the exemplar distance problem on two genomes $(G_1)$ and $(G_2)$, where each gene occurs at most $(s)$ times in $(G_1)$ and at most $(t)$ times in $(G_2)$. We show that the simplest nontrivial variant of the exemplar distance problem, $((1,2))$-Exemplar Distance, is already hard to approximate for a wide variety of distance measures, including both popular genome rearrangement measures such as adjacency disruptions, signed reversals, and signed double-cut-and-joins, and classic string edit distance measures such as Levenshtein and Hamming distances.