Unique perfect phylogeny is intractable

A phylogeny is a tree capturing evolution and ancestral relationships of a set of taxa (e.g., species). Reconstructing phylogenies from molecular data plays an important role in many areas of contemporary biological research. A phylogeny is perfect if (in rough terms) it correctly captures all input data. Determining if a perfect phylogeny exists was shown to be intractable in 1992 by Mike Steel [32] and independently by Bodlaender et al. [4]. In light of this, a related problem was proposed in [32]: given a perfect phylogeny, determine if it is the unique perfect phylogeny for the given dataset, where the dataset is provided as a set of quartet (4-leaf) trees. It was suggested that this problem may be more tractable [32], and determining its complexity became known as the Quartet Challenge [33]. In this paper, we resolve this question by showing that the problem is CoNP-complete. We prove this by relating perfect phylogenies to satisfying assignments of Boolean formulas. To this end, we cast the question as a chordal sandwich problem. As a particular consequence of our method, we show that the unique minimal chordal sandwich problem is CoNP-complete, and counting minimal chordal sandwiches is #P-complete.