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 (MaxRTC) and the minimum rooted triplets inconsistency problem (MinRTI) in which the input is a set R of rooted triplets, and where the objectives are to find a largest cardinality subset of R which is consistent and a smallest cardinality subset of R whose removal from R results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic  results in a bottom-up-based 3-approximation for MaxRTC. We then demonstrate how any approximation algorithm for MinRTI could be used to approximate MaxRTC, and thus obtain the first polynomial-time approximation algorithm for MaxRTC 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 MaxRTC and MinRTI are NP-hard even if restricted to minimally dense instances. Finally, we prove that MinRTI cannot be approximated within a ratio of O(log n) in polynomial time, unless P = NP.
|Title of host publication||Algorithms and Computation (Proceedings 19th International Symposium, ISAAC 2008, Gold Coast, Australia, December 15-17, 2008)|
|Editors||S.H. Hong, H. Nagamochi, T. Fukunaga|
|Place of Publication||Berlin|
|Publication status||Published - 2008|
|Name||Lecture Notes in Computer Science|
Byrka, J., Guillemot, S., & Jansson, J. (2008). New results on optimizing rooted triplets consistency. In S. H. Hong, H. Nagamochi, & T. Fukunaga (Eds.), Algorithms and Computation (Proceedings 19th International Symposium, ISAAC 2008, Gold Coast, Australia, December 15-17, 2008) (pp. 484-495). (Lecture Notes in Computer Science; Vol. 5369). Berlin: Springer. https://doi.org/10.1007/978-3-540-92182-0_44