Fixed-parameter tractability of the maximum agreement supertree problem

Using Robinson-Foulds supertrees in divide-and-conquer phylogeny estimation

One of the Grand Challenges in Science is the construction of the Tree of Life, an evolutionary tree containing several million species, spanning all life on earth. However, the construction of the Tree of Life is enormously computationally challenging, as all the current most accurate methods are either heuristics for NP-hard optimization problems or Bayesian MCMC methods that sample from tree space. One of the most promising approaches for improving scalability and accuracy for phylogeny estimation uses divide-and-conquer: a set of species is divided into overlapping subsets, trees are constructed on the subsets, and then merged together using a "supertree method". Here, we present Exact-RFS-2, the first polynomial-time algorithm to find an optimal supertree of two trees, using the Robinson-Foulds Supertree (RFS) criterion (a major approach in supertree estimation that is related to maximum likelihood supertrees), and we prove that finding the RFS of three input trees is NP-hard. Exact-RFS-2 is available in open source form on Github at https://github.com/yuxilin51/GreedyRFS .

COSPEDTree: COuplet Supertree by Equivalence Partitioning of Taxa Set and DAG Formation

From a set of phylogenetic trees with overlapping taxa set, a supertree exhibits evolutionary relationships among all input taxa. The key is to resolve the contradictory relationships with respect to input trees, between individual taxa subsets. Formulation of this NP hard problem employs either local search heuristics to reduce tree search space, or resolves the conflicts with respect to fixed or varying size subtree level decompositions. Different approximation techniques produce supertrees with considerable performance variations. Moreover, the majority of the algorithms involve high computational complexity, thus not suitable for use on large biological data sets. Current study presents COSPEDTree, a novel method for supertree construction. The technique resolves source tree conflicts by analyzing couplet (taxa pair) relationships for each source trees. Subsequently, individual taxa pairs are resolved with a single relation. To prioritize the consensus relations among individual taxa pairs for resolving them, greedy scoring is employed to assign higher score values for the consensus relations among a taxa pair. Selected set of relations resolving individual taxa pairs is subsequently used to construct a directed acyclic graph (DAG). Vertices of DAG represents a taxa subset inferred from the same speciation event. Thus, COSPEDTree can generate non-binary supertrees as well. Depth first traversal on this DAG yields final supertree. According to the performance metrics on branch dissimilarities (such as FP, FN and RF), COSPEDTree produces mostly conservative, well resolved supertrees. Specifically, RF metrics are mostly lower compared to the reference approaches, and FP values are lower apart from only strictly conservative (or veto) approaches. COSPEDTree has worst case time and space complexities of cubic and quadratic order, respectively, better or comparable to the reference approaches. Such high performance and low computational costs enable COSPEDTree to be applied on large scale biological data sets.

Enumerating all maximal frequent subtrees in collections of phylogenetic trees

Background

A common problem in phylogenetic analysis is to identify frequent patterns in a collection of phylogenetic trees. The goal is, roughly, to find a subset of the species (taxa) on which all or some significant subset of the trees agree. One popular method to do so is through maximum agreement subtrees (MASTs). MASTs are also used, among other things, as a metric for comparing phylogenetic trees, computing congruence indices and to identify horizontal gene transfer events.

Results

We give algorithms and experimental results for two approaches to identify common patterns in a collection of phylogenetic trees, one based on agreement subtrees, called maximal agreement subtrees, the other on frequent subtrees, called maximal frequent subtrees. These approaches can return subtrees on larger sets of taxa than MASTs, and can reveal new common phylogenetic relationships not present in either MASTs or the majority rule tree (a popular consensus method). Our current implementation is available on the web at https://code.google.com/p/mfst-miner/.

Conclusions

Our computational results confirm that maximal agreement subtrees and all maximal frequent subtrees can reveal a more complete phylogenetic picture of the common patterns in collections of phylogenetic trees than maximum agreement subtrees; they are also often more resolved than the majority rule tree. Further, our experiments show that enumerating maximal frequent subtrees is considerably more practical than enumerating ordinary (not necessarily maximal) frequent subtrees.