On the inapproximability of the exemplar conserved interval distance problem of genomes

Abstract

In this paper we present two main results about the inapproximability of the exemplar conserved interval distance problem of genomes. First, we prove that it is NP-complete to decide whether the exemplar conserved interval distance between any two genomes is zero or not. This result implies that the exemplar conserved interval distance problem does not admit any approximation in polynomial time, unless P=NP. In fact, this result holds, even when every gene appears in each of the given genomes at most three times. Second, we strengthen the first result under a weaker definition of approximation, called weak approximation. We show that the exemplar conserved interval distance problem does not admit any weak approximation within a super-linear factor of \(\frac{2}{7}m^{1.5}\) , where m is the maximal length of the given genomes. We also investigate polynomial time algorithms for solving the exemplar conserved interval distance problem when certain constrains are given. We prove that the zero exemplar conserved interval distance problem of two genomes is decidable in polynomial time when one genome is O(log n)-spanned. We also prove that one can solve the constant-sized exemplar conserved interval distance problem in polynomial time, provided that one genome is trivial.