A tutorial on the balanced minimum evolution problem

The Balanced Minimum Evolution Problem (BMEP) is an APX-hard network design problem that consists of finding a minimum length unrooted binary tree (also called a phylogeny) having as a leaf-set a given set of molecular sequences. The optimal solution to the BMEP (i.e., the optimal phylogeny) encodes the hierarchical evolutionary relationships of the input sequences. This information is crucial for a multitude of research fields, ranging from systematics to medical research, passing through drug discovery, epidemiology, ecology, biodiversity assessment and population dynamics. In this article, we introduce the reader to the problem and present the current state-of-the-art; we include the most important achievements reached so far and the challenges that still remain to be addressed.