Embedding phylogenetic trees in networks of low treewidth

TL;DR

Tree Containment asks whether a rooted binary tree $T$ embeds in a rooted binary network $N$. The authors develop a dynamic programming approach on a tree decomposition of the display graph $D_{ ext{in}}(N,T)$, introducing compact signatures and reconciliations to bound the state space and ensure correctness. They prove an explicit fixed-parameter tractable running time of $2^{O\big(tw( N_{\text{in}})^2\big)}\cdot |A(N_{\text{in}})|$, establishing the first constructive FPT algorithm parameterized by treewidth for Tree Containment. This work not only provides a practical tool for validating inferred phylogenetic networks but also lays groundwork for extending treewidth-based methods to related problems such as Network Containment and Hybridization Number, with potential broader applicability in reconciling multiple related graphs.

Abstract

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.

Figures (10)

• Figure 1: Left: a phylogenetic tree $T$. Middle: a phylogenetic network $N$ displaying $T$ (solid lines indicate an embedding of $T$; black nodes indicate reticulations). Right: the display graph $D(N,T)$ of $N$ and $T$ (see \ref{['sec:chal']}) with the network part drawn on top and the tree part drawn on the bottom. Note that vertices of the display graph are not labelled. In the figure, the leaves (square vertices) are ordered in the same way as in $N$.
• Figure 2: Left: An example of a display graph $D_\textsc{in}(N_{\textsc{in}}, T_{\textsc{in}})$ for which $N_{\textsc{in}}$ displays $T_{\textsc{in}}$ as witnessed by the embedding function $\phi$ that is indicated by bold edges. Highlighting with dashed border represents the sets $P$ and $F$, for some bag $(P,S,F)$ in a tree decomposition of $D_\textsc{in}(N_{\textsc{in}}, T_{\textsc{in}})$. Right: A representation of the (compact) signature for $(P,S,F)$ derived from this solution. Vertices labelled $\textsc{past}\xspace$ or $\textsc{future}\xspace$ are highlighted in gray without border.
• Figure 3: Example of a signature of a bag $(P,S,F)$. The $S$-part of $D(N_{\textsc{in}},T_{\textsc{in}})$ is solid while the non-$S$ part is faded. The embedding $\phi$ (right, indicated with gray edge-highlight) maps $T$ into $N$. The dotted arcs labelled $\iota$ show the isomorphism between part of $D(N,T)$ and $S\subseteq V(D(N_{\textsc{in}},T_{\textsc{in}}))$. Note that the part of $D(N,T)$ that is not mapped to $S$ is not necessarily isomorphic to anything in $D(N_{\textsc{in}},T_{\textsc{in}})$.
• Figure 4: Illustration of \ref{['def:restrict']} (except non-$\mathcal{Y}$ parts of $\iota$ and $\iota'$). Left: The example containment structure $\psi$ of \ref{['fig:containment struct']}. The dashed area indicates $S'$ and $g$: for all $u$ in the dashed area, $g(\iota(u))=\textsc{past}\xspace$ while $\iota(u)\notin\{\textsc{past}\xspace,\textsc{future}\xspace\}$. Right: The $g$-restriction of $\psi$. In accordance with \ref{['def:restrict']}, note how (a) arcs in the "past"-area of $N$ that are not mapped to by $\phi$ disappear in $N'$ while arcs that are mapped by $\phi$ persist, (b) many of the arcs in $T$ are in the "past"-area, but embedded by $\phi$ into paths of $N$ that live (at least partially) in the "future"-area of $N$ and, therefore, also persist, and (c) a common leaf of $T$ and $N$ has been removed since all its incoming arcs have been deleted. Finally, as will be the case in the dynamic programming later on, the "past"- and "future"-areas never touch in $D(N',T')$.
• Figure 5: Illustration of proof for Introduce bags, validity of $\sigma$ implies validity of $\sigma'$. Solid lines are restriction relations that we may assume; the dashed line shows the construction of $\psi'$ from $\psi$; dotted lines are relations we can infer using transitivity.

Theorems & Definitions (83)

• Definition 1: display graph
• Definition 2: embedding function
• Lemma 1
• proof
• Theorem 1: Janssen2019Treewidth
• Definition 3: isolabelling
• Definition 4: containment structure
• Definition 5: signature
• Lemma 2
• proof