Level-k Phylogenetic Networks Are Constructable from a Dense Triplet Set in Polynomial Time
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This paper considers the problem of determining whether a given set $\T$ of rooted triplets can be merged without conflicts into a galled phylogenetic network and, if so, constructing such a network. When the input $\T$ is dense, we solve the problem in $O(|\T|)$ time, which is optimal since the size of the input is $\Theta(|\T|)$. In comparison, the previously fastest algorithm for this problem runs in $O(|\T|^2)$ time. We also develop an optimal $O(|\T|)$-time algorithm for enumerating all simple phylogenetic networks leaf-labeled by $L$ that are consistent with $\T$, where $L$ is the set of leaf labels in $\T$, which is used by our main algorithm. Next, we prove that the problem becomes NP-hard if extended to nondense inputs, even for the special case of simple phylogenetic networks. We also show that for every positive integer $n$, there exists some set $\T$ of rooted triplets on $n$ leaves such that any galled network can be consistent with at most $0.4883 \cdot |\T|$ of the rooted triplets in $\T$. On the other hand, we provide a polynomial-time approximation algorithm that always outputs a galled network consistent with at least a factor of $\frac{5}{12}$ ($ 0.4166$) of the rooted triplets in $\T$.