Evolutionary Trees can be Learned in Polynomial Time in the Two-State General Markov Model

The j-State General Markov Model of evolution (due to Steel) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the Two-State General Markov Model of evolution generalises the well-known Cavender-Farris-Neyman model of evolution by removing the symmetry restriction (which requires that the probability that a ‘0’ turns into a ‘1’ along an edge is the same as the probability that a ‘1’ turns into a ‘0’ along the edge). Farach and Kannan showed how to PAC-learn Markov Evolutionary Trees in the Cavender-FarrisNeyman model provided that the target tree satisfies the additional restriction that all pairs of leaves have a sufficiently high probability of being the same. We show how to remove both restrictions and thereby obtain the first polynomial-time PAC-learning algorithm (in the sense of Kearns et al.) for the general class of Two-State Markov Evolutionary Trees.