On the approximability of the exemplar adjacency number problem for genomes with gene repetitions

In this paper, we apply a measure, exemplar adjacency number, which complements and extends the well-studied breakpoint distance between two permutations, to measure the similarity between two genomes (or in general, between any two sequences drawn from the same alphabet). For two genomes G andH drawn from the same set of n gene families and containing gene repetitions, we consider the corresponding Exemplar Adjacency Number problem (EAN), in which we delete duplicated genes from G and H such that the resultant exemplar genomes (permutations) G and H have the maximum adjacency number. We obtain the following results. First, we prove that the one-sided 2-repetitive EAN problem, i.e., when one of G and H is given exemplar and each gene occurs in the other genome at most twice, can be ∗Corresponding Author. Email addresses: [email protected] (Zhixiang Chen), [email protected] (Bin Fu), [email protected] (Randy Goebel), [email protected] (Guohui Lin), [email protected] (Weitian Tong), [email protected] (Jinhui Xu), [email protected] (Boting Yang), [email protected] (Zhiyu Zhao), [email protected] (Binhai Zhu) Preprint submitted to Theoretical Computer Science July 7, 2014 linearly reduced from the Maximum Independent Set problem. This implies that EAN does not admit any O(n)-approximation algorithm, for any ǫ > 0, unless P = NP. This hardness result also implies that EAN, parameterized by the optimal solution value, is W[1]-hard. Secondly, we show that the two-sided 2-repetitive EAN problem has an O(n)-approximation algorithm, which is tight up to a constant factor.