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Phylogenetic networks are leaf-labeled, rooted, acyclic, and directed graphs that are used to model reticulate evolutionary histories. Several measures for quantifying the topological dissimilarity between two phylogenetic networks have been devised, each of which was proven to be a metric on certain restricted classes of phylogenetic networks. A biologically motivated class of phylogenetic networks, namely, reduced phylogenetic networks, was recently introduced. None of the existing measures is a metric on the space of reduced phylogenetic networks. In this paper, we provide a metric on the space of reduced phylogenetic networks that is computable in time polynomial in the size of the networks.

Phylogenies-the evolutionary histories of groups of organisms-play a major role in representing the interrelationships among biological entities. Many methods for reconstructing and studying such phylogenies have been proposed, almost all of which assume that the underlying history of a given set of species can be represented by a binary tree. Although many biological processes can be effectively modeled and summarized in this fashion, others cannot: recombination, hybrid speciation, and horizontal gene transfer result in networks of relationships rather than trees of relationships. In previous works, we formulated a maximum parsimony (MP) criterion for reconstructing and evaluating phylogenetic networks, and demonstrated its quality on biological as well as synthetic data sets. In this paper, we provide further theoretical results as well as a very fast heuristic algorithm for the MP criterion of phylogenetic networks. In particular, we provide a novel combinatorial definition of phylogenetic networks in terms of "forbidden cycles," and provide detailed hardness and hardness of approximation proofs for the "small" MP problem. We demonstrate the performance of our heuristic in terms of time and accuracy on both biological and synthetic data sets. Finally, we explain the difference between our model and a similar one formulated by Nguyen et al., and describe the implications of this difference on the hardness and approximation results.

Phylogenies - the evolutionary histories of groups of organisms - are one of the most widely used tools throughout the life sciences, as well as objects of research within systematics, evolutionary biology, epidemiology, etc. Almost every tool devised to date to reconstruct phylogenies produces trees; yet it is widely understood and accepted that trees oversimplify the evolutionary histories of many groups of organims, most prominently bacteria (because of horizontal gene transfer) and plants (because of hybrid speciation). Various methods and criteria have been introduced for phylogenetic tree reconstruction. Parsimony is one of the most widely used and studied criteria, and various accurate and efficient heuristics for reconstructing trees based on parsimony have been devised. Jotun Hein suggested a straightforward extension of the parsimony criterion to phylogenetic networks. In this paper we formalize this concept, and provide the first experimental study of the quality of parsimony as a criterion for constructing and evaluating phylogenetic networks. Our results show that, when extended to phylogenetic networks, the parsimony criterion produces promising results. In a great majority of the cases in our experiments, the parsimony criterion accurately predicts the numbers and placements of non-tree events.

Phylogenetic networks model the evolutionary history of sets of organisms when events such as hybrid speciation and horizontal gene transfer occur. In spite of their widely acknowledged importance in evolutionary biology, phylogenetic networks have so far been studied mostly for specific data sets. We present a general definition of phylogenetic networks in terms of directed acyclic graphs (DAGs) and a set of conditions. Further, we distinguish between model networks and reconstructible ones and characterize the effect of extinction and taxon sampling on the reconstructibility of the network. Simulation studies are a standard technique for assessing the performance of phylogenetic methods. A main step in such studies entails quantifying the topological error between the model and inferred phylogenies. While many measures of tree topological accuracy have been proposed, none exist for phylogenetic networks. Previously, we proposed the first such measure, which applied only to a restricted class of networks. In this paper, we extend that measure to apply to all networks, and prove that it is a metric on the space of phylogenetic networks. Our results allow for the systematic study of existing network methods, and for the design of new accurate ones.