On Computing the Maximum Parsimony Score of a Phylogenetic Network

Abstract

Abstract. Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as Maximum Parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of Maximum Parsimony on networks, "hardwired" and "softwired", and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multicut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and softwired parsimony score can be computed efficiently using Integer Linear Programming. The software has been made freely available.

Exactly Computing the Parsimony Scores on Phylogenetic Networks Using Dynamic Programming

“…If the edges are those of a spanning tree of the network that minimizes the parsimony score, the network score assumes a treelike evolutionary scenario for the character; and reticulate events are explained by different spanning trees in the network that achieve the best parsimony score for various characters. The former is called the hardwired approach, and the latter the softwired approach (Fisher et al, 2013). In this article, we use the same terms when referring to these approaches (see Section 2.3).…”

“…This problem has received attention for many years (Schrijver, 2003, 254 for detailed references) and has been shown to be NP-complete (Dahlhaus et al, 1994) for more than two states. Recently, it has been shown to be fixed-parameter tractable with the unknown score as the parameter (Fisher et al, 2013). On the other hand, the softwired problem is a bit more challenging.…”

“…On the other hand, the softwired problem is a bit more challenging. It is proven to be NP-hard even for two states; and no fixed-parameter tractable algorithm with the unknown score as the parameter can be devised unless P = NP (Fisher et al, 2013).…”

“…In this article, we use maximum parsimony to score phylogenetic networks based on the minimum number of state changes across a subset of edges of the network for each character that are required for a given set of characters to realize the input states at the leaves of the networks. Two different criteria that extend the notion of parsimony to phylogenetic networks have been defined (Fisher et al, 2013;Kannan and Wheeler, 2012;Nakhleh et al, 2005), and attempts have been made to solve them. In this article, we will present a dynamic programming approach, generalizing Sankoff's algorithm for phylogenetic trees (Sankoff, 1975;Sankoff and Rousseau, 1975), to solve the maximum parsimony problems in networks.…”

Phylogenetic Networks that Display a Tree Twice

“…An example of when one needs to have a phylogenetic tree displayed more than once in a phylogenetic network is in the case where a phylogenetic tree is sufficient for the representation of a species' evolution but there is some uncertainty about how the present-day descendants are related to one another or how they evolved from their ancestors; hence a phylogenetic network can be used in the absence of reticulation events, when there is some uncertainty in the true (tree-shaped) phylogeny [21]. If there are a number of plausible ways in which the present-day descendants could be related to each other (represented by a number of distinct phylogenetic trees) then all the information contained in those phylogenetic trees can be held in a single phylogenetic network.…”