Efficient FPT algorithms for (strict) compatibility of unrooted phylogenetic trees

TL;DR

This paper introduces efficient fixed-parameter tractable algorithms for determining compatibility and strict compatibility of unrooted phylogenetic trees, improving computational feasibility in constructing supertrees from multiple input trees.

Contribution

The paper presents the first explicit dynamic programming algorithms for compatibility problems in unrooted phylogenetic trees with a runtime of 2^{O(k^2)}·n, reducing previous algorithmic complexity.

Findings

Algorithms run in 2^{O(k^2)}·n time.

First explicit dynamic programming solutions for these problems.

Improved computational efficiency for supertree construction.

Abstract

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 $2^{O(k^2)} \cdot n$, where $n$ is the total size of the input.