In phylogenetics, a central problem is to infer the evolutionary
relationships between a set of species X; these relationships are often
depicted via a phylogenetic tree -- a tree having its leaves univocally labeled
by elements of X and without degree-2 nodes -- called the "species tree". One
common approach for reconstructing a species tree consists in first
constructing several phylogenetic trees from primary data (e.g. DNA sequences
originating from some species in X), and then constructing a single
phylogenetic tree maximizing the "concordance" with the input trees. The
so-obtained tree is our estimation of the species tree and, when the input
trees are defined on overlapping -- but not identical -- sets of labels, is
called "supertree". In this paper, we focus on two problems that are central
when combining phylogenetic trees into a supertree: the compatibility and the
strict compatibility problems for unrooted phylogenetic trees. These problems
are strongly related, respectively, to the notions of "containing as a minor"
and "containing as a topological minor" in the graph community. Both problems
are known to be fixed-parameter tractable in the number of input trees k, by
using their expressibility in Monadic Second Order Logic and a reduction to
graphs of bounded treewidth. Motivated by the fact that the dependency on k
of these algorithms is prohibitively large, we give the first explicit dynamic
programming algorithms for solving these problems, both running in time
2O(k2)â‹…n, where n is the total size of the input.Comment: 18 pages, 1 figur