A note on convex characters, Fibonacci numbers and exponential-time algorithms

TL;DR

This paper explores convex characters in phylogenetic trees, revealing Fibonacci number connections, and introduces efficient algorithms for NP-hard problems like maximum parsimony distance and tree bisection-reconnection distance.

Contribution

It proves the number of convex characters equals Fibonacci numbers, analyzes topological neutrality for certain counts, and develops efficient algorithms for complex phylogenetic problems.

Findings

Number of convex characters equals Fibonacci number (2n-1)th.

g_2(T) equals Fibonacci number (n-1)th, independent of topology.

New algorithms for NP-hard phylogenetic problems with exponential time complexity.

Abstract

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. Given an unrooted, binary phylogenetic tree T on a set of n >= 2 taxa, a closed (but fairly opaque) expression for the number of convex characters on T has been known since 1992, and this is independent of the exact topology of T. In this note we prove that this number is actually equal to the (2n-1)th Fibonacci number. Next, we define g_k(T) to be the number of convex characters on T in which each state appears on at least k taxa. We show that, somewhat curiously, g_2(T) is also independent of the topology of T, and is equal to to the (n-1)th Fibonacci number. As we demonstrate, this topological neutrality subsequently breaks down for k >= 3. However, we show that for each fixed k >= 1, g_k(T) can be computed in O(n) time and the set of characters thus counted can be efficiently listed and sampled. We use these insights to give a simple but effective exact algorithm for the NP-hard maximum parsimony distance problem that runs in time $\Theta( \phi^{n} \cdot n^2 )$, where $\phi \approx 1.618...$ is the golden ratio, and an exact algorithm which computes the tree bisection and reconnection distance (equivalently, a maximum agreement forest) in time $\Theta( \phi^{2n}\cdot \text{poly}(n))$, where $\phi^2 \approx 2.619$.