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 (Steel, 1992) [32] and independently by Bodlaender et al. (Bodlaender et al., 1992) [4]. In light of this, a related problem was proposed by Steel (1992) [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 (Steel, 1992) [32], and determining its complexity became known as the Quartet Challenge (Steel, 2012) [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 formulae. 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.

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