New results on optimizing rooted triplets consistency

A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem ( ) and the minimum rooted triplets inconsistency problem ( ) in which the input is a set of rooted triplets, and where the objectives are to find a largest cardinality subset of which is consistent and a smallest cardinality subset of whose removal from results in a consistent set, respectively. We first show that a simple modification to Wu's Best-Pair-Merge-First heuristic [25] results in a bottom-up-based 3-approximation for . We then demonstrate how any approximation algorithm for could be used to approximate , and thus obtain the first polynomial-time approximation algorithm for with approximation ratio smaller than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any tree is approximately one third, and then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both and are NP-hard even if restricted to minimally dense instances. Finally, we prove that cannot be approximated within a ratio of Ω(logn) in polynomial time, unless P∈=∈NP.