# Constructing a Consensus Phylogeny from a Leaf-Removal Distance

## TL;DR

This paper introduces a new NP-hard problem of constructing a consensus phylogeny from discordant trees, providing approximation and parameterized algorithms to address discrepancies in evolutionary scenarios.

## Contribution

It defines a novel leaf-removal distance problem for consensus phylogeny, proves its NP-hardness, and offers approximation and parameterized algorithms for practical solutions.

## Key findings

- The problem is NP-hard.
- A 2-approximation algorithm is provided.
- A parameterized algorithm is developed for the max leaf removal variant.

## Abstract

Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of genes or species, and aims at finding a single tree that minimizes the leaf-removal distance to the input trees. This problem is a specific instance of the general consensus/supertree problem, widely used to combine or summarize discordant evolutionary trees. The problem we introduce is specifically tailored to address the case of discrepancies between the input trees due to the misplacement of individual taxa. Most supertree or consensus tree problems are computationally intractable, and we show that the problem we introduce is also NP-hard. We provide tractability results in form of a 2-approximation algorithm. We also introduce a variant that minimizes the maximum number $d$ of leaves that are removed from any input tree, and provide a parameterized algorithm for this problem with parameter $d$.