Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A kernel for the maximum agreement forest problem on multiple binary phylogenetic trees

Steven Kelk, Ruben Meuwese, Leo van Iersel

TL;DR

A modified version of the well-known chain reduction rule is produced in order to prove the existence of a kernel of size O( t * r * k ) where k is the natural parameter (the number of blocks) and r=min{max{k,3},t+1}}.

Abstract

The maximum agreement forest (MAF) problem in phylogenetics takes as input a set t >=2 of binary phylogenetic trees T on the same set of taxa X. It asks for a partition X into the smallest number of blocks such that the subtrees induced by these blocks are disjoint and have common topology across all the trees in T. We produce a modified version of the well-known chain reduction rule in order to prove the existence of a kernel of size O( t * r * k ) where k is the natural parameter (the number of blocks) and r=min{max{k,3},t+1}}. We prove this bound for both the unrooted and rooted version of the problem, and demonstrate that the bound r, the length to which common chains are truncated, is tight. Our results constitute the first kernels for MAF in the t > 2 regime.

A kernel for the maximum agreement forest problem on multiple binary phylogenetic trees

TL;DR

A modified version of the well-known chain reduction rule is produced in order to prove the existence of a kernel of size O( t * r * k ) where k is the natural parameter (the number of blocks) and r=min{max{k,3},t+1}}.

Abstract

The maximum agreement forest (MAF) problem in phylogenetics takes as input a set t >=2 of binary phylogenetic trees T on the same set of taxa X. It asks for a partition X into the smallest number of blocks such that the subtrees induced by these blocks are disjoint and have common topology across all the trees in T. We produce a modified version of the well-known chain reduction rule in order to prove the existence of a kernel of size O( t * r * k ) where k is the natural parameter (the number of blocks) and r=min{max{k,3},t+1}}. We prove this bound for both the unrooted and rooted version of the problem, and demonstrate that the bound r, the length to which common chains are truncated, is tight. Our results constitute the first kernels for MAF in the t > 2 regime.
Paper Structure (9 sections, 8 theorems, 6 equations, 8 figures)

This paper contains 9 sections, 8 theorems, 6 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction rules
  4. Chain reduction for multiple trees
  5. Tightness of the truncation length $r$
  6. Kernelization
  7. Rooted kernelization
  8. Future work
  9. Acknowledgements

Key Result

Lemma 1

Let $\mathcal{T}$ be a set of unrooted binary phylogenetic trees on $X$ and let $\mathcal{T}_S$ be the result of applying the subtree reduction rule. Then there exists an unrooted agreement forest of $\mathcal{T}_S$ of size at most $k$ if and only if $\mathcal{T}$ has an unrooted agreement forest of

Figures (8)

  • Figure 1: An unrooted binary phylogenetic tree $T$ on $X=\{a,\ldots,m\}$. For example, $T[\{a, b, c, d\}]$ is a pendant subtree, $(e, f, g)$ is a $3$-chain with $deg^T(\{e,f,g\})=2$ and $(j, k, l, m)$ is a $4$-chain in $T$ with $deg^T(\{j,k,l,m\})=1$.
  • Figure 2: Example of an unrooted agreement forest ${F} = \{\{a,b,c,d\}, \{e\}, \{f\} \}$ for phylogenetic trees $T$ and $T'$. This is a maximum agreeement forest: a smaller agreement forest is not possible.
  • Figure 3: A rooted binary phylogenetic tree $T$. For example, $T[\{a, b, c, d\}]$ is a pendant subtree, $(g, f, e)$ is a 3-chain and $(h, i, j, k)$ a 4-chain in $T$.
  • Figure 4: Example of a rooted agreement forest $F_r = \{\{a,b,c\}, \{d, e\} \}$ for the rooted phylogenetic trees $T$ and $T'$. This is a maximum agreement forest because an agreement forest with fewer blocks is not possible.
  • Figure 5: Example of applying the subtree reduction rule on $\mathcal{T} = \{T, T'\}$ with $S=\{c,d\}$, reducing the common pendant subtree to a single taxon $c$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • ...and 5 more