Efficiently solvable perfect phylogeny problems on binary and k-state data with missing values

Abstract

The perfect phylogeny problem is of central importance to both evolutionary biology and population genetics. Missing values are a common occurrence in both sequence and genotype data. In their presence, the problem of finding a perfect phylogeny is NP-hard, even for binary characters [24]. We extend the utility of the perfect phylogeny by introducing new efficient algorithms for broad classes of binary and multi-state data with missing values. Specifically, we address the rich data hypothesis introduced by Halperin and Karp [11] for the binary perfect phylogeny problem with missing data. We give an efficient algorithm for enumerating phylogenies compatible with characters satisfying the rich data hypothesis. This algorithm is useful for computing the probability of data with missing values under the coalescent model. In addition, we use the partition intersection (PI) graph and chordal graph theory to generalize the rich data hypothesis to multi-state characters with missing values. For a bounded number of states, k, we provide a fixed parameter tractable algorithm for the k-state perfect phylogeny problem with missing data. Our approach reduces missing data problems to problems on complete data. Finally, we characterize a commonly observed condition, an m-clique in the PI graph, under which a perfect phylogeny can be found efficiently for binary characters with missing values. We evaluate our results with extensive empirical analysis using two biologically motivated generative models of character data. © 2011 Springer-Verlag.