Optimizing tree and character compatibility across several phylogenetic trees

Given a set R of rooted phylogenetic trees on overlapping taxa, it takes polynomial time to decide whether or not there exists a rooted phylogenetic tree that is compatible with R. Since not all evolutionary histories for a set of species can be explained by a single tree, it is natural to ask for the minimum number of rooted phylogenetic trees needed such that each tree in R is compatible with at least one tree. This paper shows that it is computationally hard to compute this minimum number. In particular, if R contains rooted triples (rooted binary phylogenetic trees on three leaves), it is NP-complete to decide whether there exist two rooted phylogenetic trees such that each rooted triple in R is compatible with at least one of the two trees. Furthermore, for a set Σ of binary characters and a positive integer k, we show that to decide if there exists a set P of k rooted phylogenetic trees such that each character in Σ is compatible with at least one tree in P is NP-complete for all k ≥ 3, but solvable in polynomial time for k = 2. This generalizes the result for k = 1, where it is well-known to be polynomial time.