Maximal Strip Recovery Problem with Gaps: Hardness and Approximation Algorithms

Abstract

Abstract. Given two comparative maps, that is two sequences of markers each representing a genome, 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 dening a robust set of synteny blocks between dierent 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 δ-gap-MSR, where δ 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 δ-gap-MSR is NP-complete for any δ ≥ 1, and even APX-hard for any δ ≥ 2. We also provide two approximation algorithms, with ratio 1.8 for δ = 1, and ratio 4 for δ ≥ 2.

Tractability and Approximability of Maximal Strip Recovery

“…In this section, we present a (d + 1.5)-approximation algorithm for the two minimization problems CMSR-d and δ-gap-CMSR-d. Recall that 2d-approximation algorithms [2,8,1] were known for the two maximization problems MSR-d and δ-gap-MSR-d. Let k be the number of deleted markers in an optimal solution. Then the number of single-markers in the input maps is at most (2d + 1)k because each single-marker is either deleted or adjacent to a deleted marker.…”

“…For the four variants of the maximal strip recovery problem, MSR-d, CMSRd, δ-gap-MSR-d, and δ-gap-CMSR-d, several hardness results have been obtained [2,9,6,1,7,8], and a variety of algorithms have been developed, including heuristics [10], approximation algorithms [2,1,5], and FPT algorithms [9,5]. For example, it is known that MSR-d admits a 2d-approximation algorithm for any d ≥ 2 [2,8], and that δ-gap-MSR-d admits a 2d-approximation algorithm for any d ≥ 2 and δ ≥ 1 and a 1.8-approximation algorithm for d = 2 and δ = 1 [1].…”

“…Compared to the approximation upper bound of 2d [2,1,8] for the two maximization problems MSR-d and δ-gap-MSR-d, which almost matches (at least asymptotically) the current best lower bound of Ω(d/ log d) [8], our upper bound of d + 1.5 for the two minimization problems CMSR-d and δ-gap-CMSR-d is still far away from the constant lower bound in [8]. It is an intriguing question whether CMSR-d and δ-gap-CMSR-d admit approximation algorithms with constant ratios independent of d.…”

Inapproximability of maximal strip recovery

“…The idea is actually quite simple and has been used many times previously [21,19,10]. Note that any strip of length l > 3 is a concatenation of shorter strips of lengths 2 and 3, for example, 4 = 2 + 2, 5 = 2 + 3, etc.…”

“…Bulteau, Fertin, and Rusu [10] recently proposed a restricted variant of Maximal Strip Recovery called δ-gap-MSR, which is MSR-2 with the additional constraint that at most δ markers may be deleted between any two adjacent markers of a strip in each genomic map. We now define δ-gap-MSR-d and δ-gap-CMSR-d as the restricted variants of the two problems MSR-d and CMSR-d, respectively, with the additional δgap constraint.…”