Checking Phylogenetics Decisiveness in Theory and in Practice.

Suppose we aim to build a phylogeny for a set of taxa X using information from a collection of loci, where each locus offers information for only a fraction of the taxa. The question is whether, based solely on the pattern of data availability, called a taxon coverage pattern, one can determine if the data suffices to construct a reliable phylogeny. The problem can be expressed combinatorially as follows. Let us call a taxon coverage pattern decisive if, for any binary phylogenetic tree T for X, the collection of phylogenetic trees obtained by restricting T to the subset of X covered by each locus uniquely determines T. Here we relate the problem of checking whether a taxon coverage pattern is decisive to a hypergraph coloring problem. Using this connection, we (1) show that checking decisiveness is co-NP complete; (2) obtain lower bounds on the amount of coverage needed to achieve decisiveness; (3) devise an exact algorithm for decisiveness; (4) develop problem reduction rules, and use them to obtain efficient algorithms for inputs with few loci; and (5) devise Boolean satisfiability (SAT) and integer linear programming formulations (ILP) of decisiveness that allow us to analyze data sets that arise in practice. For data sets that are not decisive, we use our SAT and ILP formulations to obtain decisive subsets of the data.