Influence of Tree Topology Restrictions on the Complexity of Haplotyping with Missing Data

Haplotyping, also known as haplotype phase prediction, is the problem of predicting likely haplotypes based on genotype data. One fast haplotyping method is based on an evolutionary model where a perfect phylogenetic tree is sought that explains the observed data. Unfortunately, when data entries are missing, as is often the case in real laboratory data, the resulting formal problem IPPH, which stands for incomplete perfect phylogeny haplotyping, is NP-complete and no theoretical results are known concerning its approximability, fixed-parameter tractability or exact algorithms for it. Even radically simplified versions, such as the restriction to phylogenetic trees consisting of just two directed paths from a given root, are still NP-complete, but here, at least, a fixed-parameter algorithm is known. We generalize this algorithm to arbitrary tree topologies and present the first theoretical analysis of an algorithm that works on arbitrary instances of the original IPPH problem. At the same time we also show that restricting the tree topology does not always make finding phylogenies easier: while the incomplete directed perfect phylogeny problem is well-known to be solvable in polynomial time, we show that the same problem restricted to path topologies is NP-complete.