The approximability of maximum rooted triplets consistency with fan triplets and forbidden triplets

The NP-hard maximum rooted resolved triplets consistency problem (MRTC) takes as input a set [Formula presented] of leaf labels and a set [Formula presented] of resolved triplets over [Formula presented] and asks for a rooted phylogenetic tree that is consistent with the maximum number of elements in [Formula presented]. This article studies the approximability of a generalization of the problem called the maximum rooted triplets consistency problem (MTC) where in addition to resolved triplets, the input may contain fan triplets, forbidden resolved triplets, and forbidden fan triplets. To begin with, we observe that MTC admits a [Formula presented]-approximation in polynomial time. Next, we generalize Wu's exact exponential-time algorithm for MRTC (Wu, 2004) to MTC. Forcing the algorithm to always output a rooted [Formula presented]-ary phylogenetic tree for any specified [Formula presented] subsequently leads to an exponential-time approximation scheme (ETAS) for MTC. We then present a polynomial-time approximation scheme (PTAS) for complete instances of MTC (meaning that for every [Formula presented] with [Formula presented], [Formula presented] contains at least one rooted triplet involving the leaf labels in [Formula presented]), based on the techniques introduced by Jiang et al. (2001) for a related problem. We also study the computational complexity of MTC restricted to fan triplets and forbidden fan triplets. Finally, extensions to weighted instances are considered