Finding Maximum Common Contractions Between Phylogenetic Networks

TL;DR

This work generalizes the Robinson-Foulds paradigm to phylogenetic networks by using edge contractions and expansions to define a contraction-expansion distance $d_{ ext{CE}}$ and an MCC-based dissimilarity $oldsymbol{igtriangleup}_{ ext{MCC}}$, showing both connectivity of the network space and the strongest computational results for MCC. It establishes that $d_{ ext{CE}}$ is a true metric, while $oldsymbol{igtriangleup}_{ ext{MCC}}$ is a semi-metric, and proves MCC is NP-hard even under tight restrictions, with ETH-based hardness. To counter the hardness, the authors introduce weakly galled trees and derive a polynomial-time dynamic programming algorithm using witness structures, 1-clades, and 2-clades, achieving $O(n^5)$ time for MCC-like computation in this class. The results illuminate when a common underlying structure can be efficiently extracted from networks and provide a practical method for comparing complex evolutionary histories, with implications for bioinformatics workflows that rely on network-level comparisons.

Abstract

In this paper, we lay the groundwork on the comparison of phylogenetic networks based on edge contractions and expansions as edit operations, as originally proposed by Robinson and Foulds to compare trees. We prove that these operations connect the space of all phylogenetic networks on the same set of leaves, even if we forbid contractions that create cycles. This allows to define an operational distance on this space, as the minimum number of contractions and expansions required to transform one network into another. We highlight the difference between this distance and the computation of the maximum common contraction between two networks. Given its ability to outline a common structure between them, which can provide valuable biological insights, we study the algorithmic aspects of the latter. We first prove that computing a maximum common contraction between two networks is NP-hard, even when the maximum degree, the size of the common contraction, or the number of leaves is bounded. We also provide lower bounds to the problem based on the Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time algorithm for weakly-galled trees, a generalization of galled trees.

Figures (10)

• Figure 1: (top) The transformation of a network into another through contractions, followed by expansions. Midway, the intermediate network is a maximum common contraction, which can be achieved from $\mathcal{N}_2$ by reversing the expansions. (bottom) Illustration of edge contractions (left) and expansions (right). For expansions, the sets $X^-,Y^-,Z^-,X^+,Y^+,Z^+$ specify how the neighbors of $u$ are distributed to $v$ and $w$. Note that contractions may delete cycles and expansions create them.
• Figure 2: (A) Examples of $3$ networks for which $\delta_\text{MCC}(\mathcal{N}_1,\mathcal{N}_3)>\delta_\text{MCC}(\mathcal{N}_1,\mathcal{N}_2) + \delta_\text{MCC}(\mathcal{N}_2, \mathcal{N}_3)$. Indeed, as shown on the figure, there is a common contraction of size $4$ between $\mathcal{N}_1$ and $\mathcal{N}_2$, and of size $5$ between $\mathcal{N}_2$ and $\mathcal{N}_3$. However, the only possible common contraction between $\mathcal{N}_1$ and $\mathcal{N}_3$ is the star network. This is due the edge highlighted in $\mathcal{N}_1$ not being admissible, enforcing the contraction of the whole cycle. Since all networks have $6$ internal nodes, we have $\delta_\text{MCC}(\mathcal{N}_1,\mathcal{N}_3)=12-2=10$ whereas $\delta_\text{MCC}(\mathcal{N}_1,\mathcal{N}_2)=12-8=4$ and $\delta_\text{MCC}(\mathcal{N}_2,\mathcal{N}_3)=12-10=2$. (B) An example further highlighting the difference between $d_{\text{CE}}$ and $\delta_\text{MCC}$, as we have there two networks $\mathcal{N}_1,\mathcal{N}_2$ such that $d_{\text{CE}}(\mathcal{N}_1,\mathcal{N}_2)=2$ (following the top path) whereas $\delta_\text{MCC}(\mathcal{N}_1,\mathcal{N}_2)=6$ (following the bottom path).
• Figure 3: (left) Two caterpillar trees over $7$ leaves whose common contraction is the star network. One way of seeing it is that in $T$, leaf $1$ is a child of the root, and therefore this must be the case in a contraction of $T$ as well. In $T'$, the only way to achieve this is to contract the parent of leaf $1$ into the root, making the star network the only possible common contraction. (right) Reticulation edges have been added in $T,T'$ between different pairs of leaves ($P_N=\{(2,3)\}$ and $P_{N'}=\{(3,4)\}$ in the notations of the proof), in order to get $N,N'$ with respectively $7$ and $9$ internal nodes, while maintaining the property that the star network is the only possible common contraction.
• Figure 4: Illustration of our first reduction from Set Splitting to Maximum Common Network Contraction on an example.
• Figure 5: Illustration of our second reduction, also from Set Splitting. The blue nodes represent the sets of $\mathcal{A}$, and the orange nodes the elements of $X$. We use this construction in Theorem \ref{['thm:hardness-p4']}, to prove that even when restricted to networks on $5$ leaves, Maximum Common Network Contraction is NP-hard.

Theorems & Definitions (43)

• Remark 1: graph isomorphism
• Definition 1: Contraction
• Proposition 1
• proof
• Definition 2: Expansion
• Definition 3: contraction-expansion distance
• Definition 4: MCC dissimilarity measure
• Remark 2
• Proposition 2
• proof