On Computing the Dollo-1 phylogeny in polynomial time

The Dollo model for reconstructing evolutionary trees from binary characters has been proposed as a generalization of the infinite sites model, also known as the Perfect Phylogeny. In particular, the Dollo model is considered more realistic than the Perfect Phylogeny for inferring the evolution of tumor mutations. In the case of binary matrices, the Dollo-$k$ model requires an evolutionary tree in which each character, corresponding to a column in the input matrix, may change from $0$ to $1$ at most once, and from $1$ to $0$ at most $k$ times throughout the entire tree. Given a binary matrix, the problem of deciding whether there exists a Dollo-$k$ tree compatible with the matrix is NP-complete for any fixed $k \geq 2$, while computing a Dollo-$0$ tree corresponds to the Perfect Phylogeny decision problem, which admits a simple linear-time algorithm. The Dollo-$1$ tree problem corresponds to the Persistent Phylogeny problem, whose computational complexity, albeit under an equivalent formulation, was posed as an open question 20 years ago. We solve this problem by presenting a polynomial-time algorithm for the Persistent Phylogeny problem. Our solution relies on efficiently solving a specific class of binary matrices, represented as bipartite graphs called \emph{skeleton graphs}, or simply skeletons. In these graphs, characters are \emph{maximal}, that is their corresponding sets of species are not related by inclusion.