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.
Cite
@article{arxiv.1704.00497,
title = {UPGMA and the normalized equidistant minimum evolution problem},
author = {Vincent Moulton and Andreas Spillner and Taoyang Wu},
journal= {arXiv preprint arXiv:1704.00497},
year = {2017}
}