  Motivated by comparative genomics, Chen et al. [9] introduced the Maximum
Duo-preservation String Mapping (MDSM) problem in which we are given two
strings $s_1$ and $s_2$ from the same alphabet and the goal is to find a
mapping $\pi$ between them so as to maximize the number of duos preserved. A
duo is any two consecutive characters in a string and it is preserved in the
mapping if its two consecutive characters in $s_1$ are mapped to same two
consecutive characters in $s_2$. The MDSM problem is known to be NP-hard and
there are approximation algorithms for this problem [3, 5, 13], but all of them
consider only the "unweighted" version of the problem in the sense that a duo
from $s_1$ is preserved by mapping to any same duo in $s_2$ regardless of their
positions in the respective strings. However, it is well-desired in comparative
genomics to find mappings that consider preserving duos that are "closer" to
each other under some distance measure [19]. In this paper, we introduce a
generalized version of the problem, called the Maximum-Weight Duo-preservation
String Mapping (MWDSM) problem that captures both duos-preservation and
duos-distance measures in the sense that mapping a duo from $s_1$ to each
preserved duo in $s_2$ has a weight, indicating the "closeness" of the two
duos. The objective of the MWDSM problem is to find a mapping so as to maximize
the total weight of preserved duos. In this paper, we give a polynomial-time
6-approximation algorithm for this problem.
