Computing a smallest multilabeled phylogenetic tree from rooted triplets.

We investigate the computational complexity of inferring a smallest possible multilabeled phylogenetic tree (MUL tree) which is consistent with each of the rooted triplets in a given set. This problem has not been studied previously in the literature. We prove that even the very restricted case of determining if there exists a MUL tree consistent with the input and having just one leaf duplication is an NP-hard problem. Furthermore, we show that the general minimization problem is difficult to approximate, although a simple polynomial-time approximation algorithm achieves an approximation ratio close to our derived inapproximability bound. Finally, we provide an exact algorithm for the problem running in exponential time and space. As a by-product, we also obtain new, strong inapproximability results for two partitioning problems on directed graphs called ACYCLIC PARTITION and ACYCLIC TREE-PARTITION.

Robustness of phylogenetic inference based on minimum evolution.

Minimum evolution is the guiding principle of an important class of distance-based phylogeny reconstruction methods, including neighbor-joining (NJ), which is the most cited tree inference algorithm to date. The minimum evolution principle involves searching for the tree with minimum length, where the length is estimated using various least-squares criteria. Since evolutionary distances cannot be known precisely but only estimated, it is important to investigate the robustness of phylogenetic reconstruction to imprecise estimates for these distances. The safety radius is a measure of this robustness: it consists of the maximum relative deviation that the input distances can have from the correct distances, without compromising the reconstruction of the correct tree structure. Answering some open questions, we here derive the safety radius of two popular minimum evolution criteria: balanced minimum evolution (BME) and minimum evolution based on ordinary least squares (OLS + ME). Whereas BME has a radius of 1/2, which is the best achievable, OLS + ME has a radius tending to 0 as the number of taxa increases. This difference may explain the gap in reconstruction accuracy observed in practice between OLS + ME and BME (which forms the basis of popular programs such as NJ and FastME).

Fixed-parameter tractability of the maximum agreement supertree problem.

Given a set L of labels and a collection of rooted trees whose leaves are bijectively labeled by some elements of L, the Maximum Agreement Supertree (SMAST) problem is given as follows: find a tree T on a largest label set L(') is included in L that homeomorphically contains every input tree restricted to L('). The problem has phylogenetic applications to infer supertrees and perform tree congruence analyses. In this paper, we focus on the parameterized complexity of this NP-hard problem, considering different combinations of parameters as well as particular cases. We show that SMAST on k rooted binary trees on a label set of size n can be solved in O((8n)k) time, which is an improvement with respect to the previously known O(n3k2) time algorithm. In this case, we also give an O((2k)pkn2) time algorithm, where p is an upper bound on the number of leaves of L missing in a SMAST solution. This shows that SMAST can be solved efficiently when the input trees are mostly congruent. Then, for the particular case where any triple of leaves is contained in at least one input tree, we give O(4pn3) and O(3:12p + n4) time algorithms, obtaining the first fixed-parameter tractable algorithms on a single parameter for this problem. We also obtain intractability results for several combinations of parameters, thus indicating that it is unlikely that fixed-parameter tractable algorithms can be found in these particular cases.

PhySIC: a veto supertree method with desirable properties.

This paper focuses on veto supertree methods; i.e., methods that aim at producing a conservative synthesis of the relationships agreed upon by all source trees. We propose desirable properties that a supertree should satisfy in this framework, namely the non-contradiction property (PC) and the induction property (PI). The former requires that the supertree does not contain relationships that contradict one or a combination of the source topologies, whereas the latter requires that all topological information contained in the supertree is present in a source tree or collectively induced by several source trees. We provide simple examples to illustrate their relevance and that allow a comparison with previously advocated properties. We show that these properties can be checked in polynomial time for any given rooted supertree. Moreover, we introduce the PhySIC method (PHYlogenetic Signal with Induction and non-Contradiction). For k input trees spanning a set of n taxa, this method produces a supertree that satisfies the above-mentioned properties in O(kn(3) + n(4)) computing time. The polytomies of the produced supertree are also tagged by labels indicating areas of conflict as well as those with insufficient overlap. As a whole, PhySIC enables the user to quickly summarize consensual information of a set of trees and localize groups of taxa for which the data require consolidation. Lastly, we illustrate the behaviour of PhySIC on primate data sets of various sizes, and propose a supertree covering 95% of all primate extant genera. The PhySIC algorithm is available at http://atgc.lirmm.fr/cgi-bin/PhySIC.