Algorithms and Hardness for Scaffold Filling to Maximize Increased Duo-Preservations

Scaffold filling is a critical step in DNA assembly. Its basic task is to fill the missing genes (fragments) into an incomplete genome (scaffold) to make it similar to the reference genome. There have been a lot of work under distinct measurements in the literature of genome comparison. For genomes with gene duplications, common string partition reveals the similarity more precisely, since it constructs a one-to-one correspondence between the same segments in the two genomes. In this paper, we adopt duo-preservation as the measurement, which is the complement of common string partition, i.e., the number of duo-preservations added to the number of common strings is exactly the length of a genome. Towards a proper scaffold filling, we just focus on the increased duo-preservations. This problem is called scaffold filling to maximize increased duo-preservations (abbr. SF-MIDP). We show that SF-MIDP is solvable in linear time for a simple version where all the genes of the scaffold are matched in a block-matching, but MAX SNP-complete for the general version, and cannot be approximated within
$\frac{16263}{16262}$
. Moreover, we present a basic approximation algorithm of factor 2, by which the optimal solution can be described in a new way, and then, improve the approximation factor to
$\frac{12}{7}$
via a greedy method.