No abstract
Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability results for the minimum temporal-hybridization number problem. This is the optimization version of the problem where we wish to construct a temporal phylogenetic network that topologically embeds two given rooted binary phylogenetic trees and that has a minimum number of indegree-2 nodes, which represent events such as hybridization and horizontal gene transfer. We end on a positive note, pointing out that fixed parameter tractability results in this area are likely to ensure the continued relevance of the temporal phylogenetic network model.
Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network N embeds as the display set of N , several questions in the context of the display set of N have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is Π P 2 -complete. Along the way, we show that two other problems are also Π P 2 -complete, one of which being a generalization of Tree-Containment.Classes of phylogenetic networks. Let N be a phylogenetic network on X with vertex set V . An edge e = (u, v) is a shortcut if there is a directed path from u to v whose set of edges does not contain e. A vertex v of N is called visible if there exists a leaf ∈ X such that each directed path from the root of N to passes through v. Now N is reticulation-visible if each reticulation in N is visible, and N is tree-child if each non-leaf vertex in N has a child that is a leaf or a tree vertex. Lastly, N is normal if it is treechild and does not contain any shortcuts. Clearly, by definition, each normal network is also tree-child. Furthermore, it follows from the next well-known equivalence result [2] that each tree-child network is also reticulation-visible.Lemma 2.1. Let N be a phylogenetic network. Then N is tree-child if and only if each vertex of N is visible.Thus, the class of normal networks is a subclass of tree-child networks. Furthermore, if there exists a map t : V → R + that assigns a time stamp to each vertex of N and satisfies the following two properties:a tree edge, then we say that N is temporal, in which case we call t a temporal labeling of N . Note that, although normal networks have no shortcuts, a normal network need not be temporal. Tree-child, normal, and temporal networks were first introduced by Cardona et al. [2], Willson [17], and Moret et al. [11], respectively.Caterpillars. Let C be a phylogenetic tree with leaf set { 1 , 2 , . . . , n }. Furthermore, for each i ∈ {1, 2...
We show N P-completeness for several planar variants of the monotone satisfiability problem with bounded variable appearances. With one exception the presented variants have an associated bipartite graph where the vertex degree is bounded by at most four. Hence, a planar and orthogonal drawing for these graphs can be computed efficiently, which may turn out to be useful in reductions using these variants as a starting point for proving some decision problem to be N P-hard.
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