Efficient FPT Algorithms for (Strict) Compatibility of Unrooted Phylogenetic Trees

Abstract

In phylogenetics, a central problem is to infer the evolutionary relationships between a set of $X$; 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 tree consists in first constructing several phylogenetic trees from primary data (e.g. DNA sequences originating from some 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 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 $2^{O(k^2)} \cdot n$, where $n$ is the total size of the input.