On Approaching the One-Sided Exemplar Adjacency Number Problem

Abstract

Given one generic linear genome $\mathcal{G}$ with gene duplications (over n gene families), an exemplar genome G is a permutation obtained from $\mathcal{G}$ by deleting duplicated genes such that G contains exactly one gene from each gene family (i.e., G is a permutation of length n). If we relax the constraint such that $G^{+}$ is obtained in the same way but has length at least k, then we call $G^{+}$ a pseudo-exemplar genome. Given $\mathcal{G}$ and one exemplar genome H over the same set of n gene families, the One-sided Exemplar Adjacency Number problem (One-sided EAN) is defined as follows: delete duplicated genes from the genome $\mathcal{G}$ to obtain an exemplar genome G of length n, such that the number of adjacencies between G and H is maximized. It is known that the problem is NP-hard; in fact, almost as hard to approximate as Independent Set, even when each gene (from the same gene family) appears at most twice in the generic genome $\mathcal{G}$. To overcome the constraint on the length of G, we define a slightly more general problem (One-sided EAN+) where we only need to obtain a pseudo-exemplar genome $G^{+}$ from $\mathcal{G}$ (by deleting duplicated genes) such that the number of adjacencies in H and $G^{+}$ is maximized. While One-sided EAN+ contains One-sided EAN as a special case, it does give us some flexibility in designing an algorithm. Firstly, we reformulate and relax the One-sided EAN+ problem as the maximum independent set (MIS) on a colored interval graph and hence reduce the appearance of each gene to at most two times. We show that this new relaxation is still NP-complete, though a simple factor-2 approximation algorithm can be designed; moreover, we also prove that the problem cannot be approximated within $2-\varepsilon$ by a local search technique. Secondly, we use integer linear programming (ILP) to solve this relaxed problem exactly. Finally, we compare our results with the up-to-date software GREDU, with various simulation data. It turns out that our algorithm is more stable and can process genomes of length up to 12,000 (while GREDU sometimes can falter on such a large dataset).