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

Abstract

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 ) · n, where n is the total size of the input.

Parameterized Algorithms in Bioinformatics: An Overview

“…More generally, using powerful meta-theorems [147,148] (problems formulatable in "Monadic Second Order Logic" (MSOL) are FPT for the treewidth of the input structure), TCY is fixed-parameter tractable for the treewidth of the display graph [149,150] (that is, the result of identifying all leaves of the same label in the disjoint union of the input trees), which is smaller than the number t of trees. Baste et al [151] improved the impractical running time resulting from the application of the meta-theorems, showing an O * (2 O(t 2 ) )-time algorithm.…”

Tree Diet: Reducing the Treewidth to Unlock FPT Algorithms in RNA Bioinformatics

“…Graph models and parameterized algorithms are found at the core of a sizable proportion of algorithmic methods in bioinformatics addressing a wide array of subfields, spanning sequence processing [40], structural bioinformatics [42], comparative genomics [7], phylogenetics [2], and further examples that can be found in a review by Bulteau and Weller [8]. RNA bioinformatics is no exception, with the prevalence of the secondary structure, an outer planar graph [39], as an abstraction of RNA conformations, and the notable utilization of graph models to represent complex topological motifs called pseudoknots [41], inducing the hardness of several tasks, such as structure prediction [1, 23, 29], structure alignment [5], or structure/sequence alignment [25].…”

“…Graph models and parameterized algorithms are found at the core of a sizable proportion of algorithmic methods in bioinformatics addressing a wide array of subfields, spanning sequence processing [40], structural bioinformatics [42], comparative genomics [7], phylogenetics [2],…”

Using Robinson-Foulds supertrees in divide-and-conquer phylogeny estimation

“…When restricting to methods that return trees on the full set of taxa, much less seems to be understood about finding supertrees on two trees. However, if the two input trees are compatible (i.e., there is a supertree that equals or refines each tree when restricted to the respective leaf set), then finding that compatibility supertree is solvable in polynomial time, using (for example) the well-known BUILD algorithm [ 20 ], but more efficient algorithms exist (e.g., [ 19 , 21 ]).…”