Better methods for solving parsimony and compatibility

Evolutionary tree reconstruction is a challenging problem with important applications in biology and linguistics, In biology, one of the most promising approaches to tree reconstruction is to find the "maximum parsimony" tree, while in linguistics, the use of the "maximum compatibility" method has been very useful. However, these problems are NP-hard, and current approaches to solving these problems amount to heuristic searches through the space of possible tree topologies (a search which can, on large trees, take months to complete), In this paper, we present a new technique, Optimal Tree Refinement, for reconstructing very large trees. Our technique is motivated by recent experimental studies which have shown that certain polynomial time methods often return contractions of the true tree. We study the use of this technique in solving maximum parsimony and maximum compatibility, and present both hardness results and polynomial time algorithms.