Perfect Path Phylogeny Haplotyping with Missing Data Is Fixed-Parameter Tractable

Haplotyping via perfect phylogeny is a method for retrieving haplotypes from genotypes. Fast algorithms are known for computing perfect phylogenies from complete and error-free input instances—these instances can be organized as a genotype matrix whose rows are the genotypes and whose columns are the single nucleotide polymorphisms under consideration. Unfortunately, in the more realistic setting of missing entries in the genotype matrix, even restricted forms of the perfect phylogeny haplotyping problem become NP-hard. We show that haplotyping via perfect phylogeny with missing data becomes computationally tractable when imposing additional biologically motivated constraints. Firstly, we focus on asking for perfect phylogenies that are paths, which is motivated by the discovery that yin-yang haplotypes span large parts of the human genome. A yin-yang haplotype implies that every corresponding perfect phylogeny has to be a path. Secondly, we assume that the number of missing entries in every column of the input genotype matrix is bounded. We show that the perfect path phylogeny haplotyping problem is fixed-parameter tractable when we consider the maximum number of missing entries per column of the genotype matrix as parameter. The restrictions we impose are met by a majority of the problem instances encountered in publicly available human genome data.