Approximating Weighted Duo-Preservation in Comparative Genomics

Motivated by comparative genomics, Chen et al. [9] introduced the Maximum Duo-preservation String Mapping (MDSM) problem in which we are given two strings s1 and s2 from the same alphabet and the goal is to find a mapping π 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 s1 are mapped to same two consecutive characters in s2. 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 s1 is preserved by mapping to any same duo in s2 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 s1 to each preserved duo in s2 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.