Shorelines of Islands of Tractability: Algorithms for Parsimony and Minimum Perfect Phylogeny Haplotyping Problems

The problem parsimony haplotyping (PH) asks for the smallest set of haplotypes that can explain a given set of genotypes, and the problem minimum perfect phylogeny haplotyping (MPPH) asks for the smallest such set that also allows the haplotypes to be embedded in a perfect phylogeny, an evolutionary tree with biologically motivated restrictions. For PH, we extend recent work by further mapping the interface between "easy" and "hard" instances, within the framework of (k, f)-bounded instances, where the number of 2s per column and row of the input matrix is restricted. By exploring, in the same way, the tractability frontier of MPPH, we provide the first concrete positive results for this problem. In addition, we construct for both PH and MPPH polynomial time approximation algorithms, based on properties of the columns of the input matrix.