Ancestral maximum likelihood of evolutionary trees is hard

Louigi Addario-Berry, Benny Chor, Mike Hallett, Jens Lagergren, Alessandro Panconesi, Todd Wareham*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Maximum likelihood (ML) (Neyman, 1971) is an increasingly popular optimality criterion or selecting evolutionary trees. Finding optimal ML trees appears to be a very hard computational task - in particular, algorithms and heuristics for ML take longer to run than algorithms and heuristics for maximum parsimony (MP). However, while MP has been known to be NP-complete for over 20 years, no such hardness result has been obtained so far for ML. In this work we make a first step in this direction by proving that ancestral maximum likelihood (AML) is NP-complete. The input to this problem is a set of aligned sequences of equal length and the goal is to find a tree and an assignment of ancestral sequences for all of that tree's internal vertices such that the likelihood of generating both the ancestral and contemporary sequences is maximized. Our NP-hardness proof follows that for MP given in (Day, Johnson and Sankoff, 1986) in that we use the same reduction from Vertex Cover; however, the proof of correctness for this reduction relative to AML is different and substantially more involved.

Original languageEnglish
Pages (from-to)257-271
Number of pages15
JournalJournal of Bioinformatics and Computational Biology
Issue number2
StatePublished - Jun 2004


  • Computational complexity
  • Maximum likelihood
  • Phylogeny inference


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