Maximum Likelihood of Evolutionary Trees is Hard

Tamir Tuller

Maximum likelihood  (ML) is an increasingly popular optimality
criterion for selecting evolutionary trees.
Finding optimal ML trees appears to be a very hard computational
task, but for tractable cases, ML is the method of choice. In
particular, algorithms and heuristics for ML take longer to run
than algorithms and heuristics for the second major character
based criterion, maximum parsimony (MP). However, while MP has
been known to be NP-complete for over 20 years, such a
hardness result for ML has so far eluded researchers in the field.

We resolve this question, and show that ML on phylogenetic trees is indeed
computationally intractable (NP hard). The crux of our work is not
the reduction (from an approximation version of
vertex cover), but its correctness proof. The proof goes through a
series of tree modifications, while controlling the likelihood
losses at each step, using quantitative
relations between parsimony values and the corresponding log
likelihood values.

This is joint work with Benny Chor.