# UPGMA and the normalized equidistant minimum evolution problem

## TL;DR

This paper demonstrates that UPGMA is a greedy heuristic for the NP-hard NEME problem, introduces approximation algorithms, and provides insights into UPGMA's behavior in phylogenetic tree construction.

## Contribution

It establishes the NP-hardness of the NEME problem and offers new heuristic and approximation algorithms, including a polynomial-time method with logarithmic approximation guarantees.

## Key findings

- UPGMA is a greedy heuristic for NEME.
- NEME problem is NP-hard.
- A polynomial-time algorithm approximates NEME within O(log^2 n).

## Abstract

UPGMA (Unweighted Pair Group Method with Arithmetic Mean) is a widely used clustering method. Here we show that UPGMA is a greedy heuristic for the normalized equidistant minimum evolution (NEME) problem, that is, finding a rooted tree that minimizes the minimum evolution score relative to the dissimilarity matrix among all rooted trees with the same leaf-set in which all leaves have the same distance to the root. We prove that the NEME problem is NP-hard. In addition, we present some heuristic and approximation algorithms for solving the NEME problem, including a polynomial time algorithm that yields a binary, rooted tree whose NEME score is within O(log^2 n) of the optimum. We expect that these results to eventually provide further insights into the behavior of the UPGMA algorithm.