The Approximability of Maximum Rooted Triplets Consistency with Fan Triplets and Forbidden Triplets

The maximum rooted resolved triplets consistency problem takes as input a set R of resolved triplets and asks for a rooted phylogenetic tree that is consistent with the maximum number of elements in R. This paper studies the polynomial-time approximability of a generalization of the problem where in addition to resolved triplets, the input may contain fan triplets and forbidden triplets. To begin with, we observe that the generalized problem admits a 1/4-approximation in polynomial time. Next, we present a polynomial-time approximation scheme (PTAS) for dense instances based on smooth polynomial integer programming. Finally, we generalize Wu’s exact exponential-time algorithm in [19] for the original problem to also allow fan triplets, forbidden resolved triplets, and forbidden fan triplets. Forcing the algorithm to always output a k-ary phylogenetic tree for any specified k ≥ 2 then leads to an exponential-time approximation scheme (ETAS) for the generalized, unrestricted problem.