Maximal Strip Recovery Problem with Gaps: Hardness and Approximation Algorithms

Given two comparative maps, that; is two sequences of markers each representing a genorne, the Maximal Strip Recovery problem (MSR) asks to extract a largest sequence of markers from each map such that the two extracted sequences are decomposable into non-overlapping strips (or synteny blocks). This aims at defining a robust set of synteny blocks between different species, which is a key to understand the evolution process since their last common ancestor. In this paper, we add a fundamental constraint to the initial problem, which expresses the biologically sustained need to bound the number of intermediate (non-selected) markers between two consecutive markers in a strip. We therefore introduce the problem delta-gap-MSR, where delta is a (usually small) non-negative integer that upper bounds the number of non-selected markers between two consecutive markers in a strip. Depending on the nature of the comparative maps (i.e., with or without duplicates), we show that delta-gap-MSR. is NP-complete for any delta >= 1, and even APX-hard for any delta >= 2. We also provide two approximation algorithms, with ratio 1.8 for delta = 1, and ratio 4 for delta >= 2.