# Minimum evolution

Minimum evolution is a distance method employed in phylogenetics modeling. It shares with maximum parsimony the aspect of searching for the phylogeny that has the shortest total sum of branch lengths.
The theoretical foundations of the minimum evolution (ME) criterion lay in the seminal works of both Kidd and Sgaramella-Zonta (1971) and Rzhetsky and Nei (1993). In these frameworks, the molecular sequences from taxa are replaced by a set of measures of their dissimilarity (i.e., the so-called "evolutionary distances") and a fundamental result states that if such distances were unbiased estimates of the true evolutionary distances from taxa (i.e., the distances that one would obtain if all the molecular data from taxa were available), then the true phylogeny of taxa would have an expected length shorter than any other possible phylogeny T compatible with those distances.

## Relationships with and comparison with other methods
### Maximum parsimony

It is worth noting here a subtle difference between the maximum-parsimony criterion and the ME criterion: while maximum-parsimony is based on an abductive heuristic, i.e., the plausibility of the simplest evolutionary hypothesis of taxa with respect to the more complex ones, the ME criterion is based on Kidd and Sgaramella-Zonta's conjectures that were proven true 22 years later by Rzhetsky and Nei. These mathematical results set the ME criterion free from the Occam's razor principle and confer it a solid theoretical and quantitative basis.
Maximum-parsimony criterion, which uses Hamming distance branch lengths, was shown to be statistically inconsistent in 1978. This led to an interest in statistically consistent alternatives such as ME.

### Neighbor joining

Neighbor joining may be viewed as a greedy heuristic for the balanced minimum evolution (BME) criterion. Saito and Nei's 1987 NJ algorithm far predates the BME criterion of 2000. For two decades, researchers used NJ without a firm theoretical basis for why it works.

## Statistical consistency

The ME criterion is known to be statistically consistent whenever the branch lengths are estimated via the Ordinary Least-Squares (OLS) or via linear programming.
However, as observed in Rzhetsky & Nei's article, the phylogeny having the minimum length under the OLS branch length estimation model may be characterized, in some circumstance, by negative branch lengths, which unfortunately are empty of biological meaning.
To solve this drawback, Pauplin proposed to replace OLS with a new particular branch length estimation model, known as balanced basic evolution (BME). Richard Desper and Olivier Gascuel showed that the BME branch length estimation model ensures the general statistical consistency of the minimum length phylogeny as well as the non-negativity of its branch lengths, whenever the estimated evolutionary distances from taxa satisfy the triangle inequality.
Le Sy Vinh and Arndt von Haeseler have shown, by means of massive and systematic simulation experiments, that the accuracy of the ME criterion under the BME branch length estimation model is by far the highest in distance methods and not inferior to those of alternative criteria based e.g., on Maximum Likelihood or Bayesian Inference. Moreover, as shown by Daniele Catanzaro, Martin Frohn and Raffaele Pesenti, the minimum length phylogeny under the BME branch length estimation model can be interpreted as the (Pareto optimal) consensus tree between concurrent minimum entropy processes encoded by a forest of n phylogenies rooted on the n analyzed taxa. This particular information theory-based interpretation is conjectured to be shared by all distance methods in phylogenetics.

## Algorithmic aspects

The "minimum evolution problem" (MEP), in which a minimum-summed-length phylogeny is derived from a set of sequences under the ME criterion, is said to be NP-hard. The "balanced minimum evolution problem" (BMEP), which uses the newer BME criterion, is APX-hard.
A number of exact algorithms solving BMEP have been described. The best known exact algorithm remains impractical for more than a dozen taxa, even with multiprocessing. There is only one approximation algorithm with proven error bounds, published in 2012.
In practical use, BMEP is overwhelmingly implemented by heuristic search. The basic, aforementioned neighbor-joining algorithm implements a greedy version of BMEP. FastME, the "state-of-the-art", starts with a rough tree then improves it using a set of topological moves such as Nearest Neighbor Interchanges (NNI). Compared to NJ, it is about as fast and more accurate. Metaheuristics have also been used.