Abstract :
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^{prime} subseteq L that homeomorphically contains every input tree restricted to L^{prime}. 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(n^{3k^{2}}) time algorithm. In this case, we also give an O((2k)^{p}kn^{2}) 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(4^{p}n^{3}) and O(3.12^{p} + n^{4}) 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.
