Incremental Haplotype Inference, Phylogeny, and Almost Bipartite Graphs∗

We address the combinatorial problem of inferring haplotypes in a population that forms a perfect phylogeny (PP) given a sample of genotypes. The problem is relevant because, in DNA sequencing, genotypes are easier to obtain than haplotyping by DNA sequencing. Since PP’s appear naturally and frequently on DNA sequences of restricted length, PP haplotyping is a favourable approach to facilitate reliable haplotype inference. Since Gusfield’s seminal paper from 2002, a number of different algorithms have been proposed. Here we give an algorithm that identifies haplotypes incrementally (along the sequence). Under the random mating assumption, all sufficiently frequent haplotypes are inferred from a random genotype sample of asymptotically optimal size. By its extreme simplicity, the idea of the algorithm easily extends to more general population structures. This can be beneficial because the strict PP assumption is easily violated in reality. Missing data can also be recovered by incremental haplotyping, if they are not too prevalent. In a more graph-theoretic part of this work we solve a problem we call almost-2-coloring of graphs, which arises in an enhanced version of our haplotyping algorithm. We show that the solution space of this graph problem can be computed in