Single nucleotide polymorphisms (SNPs) are the most frequent form of human genetic variation, of foremost importance for a variety of applications including medical diagnostic, phylogenies and drug design. The complete SNPs sequence information from each of the two copies of a given chromosome in a diploid genome is called a haplotype. The Hyplotyping Problem for a single individual is as follows: given a set of fragments from one individual's DNA, find a maximally consistent pair of SNPs haplotypes (one per chromosome copy) by removing data "errors" related to sequencing errors, repeats, and paralogous recruitment. Two versions of the problem, i.e. the Minimum Fragment Removal (MFR) and the Minimum SNP Removal (MSR), are considered. The Haplotyping Problem was introduced in [8], where it was proved that both MSR and MFR are polynomially solvable when each fragment covers a set of consecutive SNPs (i.e., it is a gapless fragment), and NP-hard in general. The original algorithms of [8] are of theoretical interest, but by no means practical. In fact, one relies on finding the maximum stable set in a perfect graph, and the other is a reduction to a network flow problem. Furthermore, the reduction does not work when there are fragments completely included in others, and neither algorithm can be generalized to deal with a bounded total number of holes in the data. In this paper, we give the first practical algorithms for the Haplotyping Problem, based on Dynamic Programming. Our algorithms do not require the fragments to not include each other, and are polynomial for each constant k bounding the total number of holes in the data. For m SNPs and n fragments, we give an O(mn~(2k+2)) algorithm for the MSR problem, and an O(2~(2k)m~2n + 2~(3k)m~3) algorithm for the MFR problem, when each fragment has at most k holes. In particular, we obtain an O(mn~2) algorithm for MSR and an (O(m~2n + m~3) algorithm for MFR on gapless fragments. Finally, we prove that both MFR and MSR are APX-hard in general.
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