Orchestrating quartets: approximation and data correction

Inferring evolutionary trees has long been a challenging problem both for biologists and computer scientists. In recent years research has concentrated on the quartet method paradigm for inferring evolutionary trees. Quartet methods proceed by first inferring the evolutionary history for every set of four species (resulting in a set Q of inferred quarter topologies) and then recombining these inferred quarter topologies to form an evolutionary tree. This paper presents two results on the quartet method paradigm. The first is a polynomial time approximation scheme (PTAS) for recombining the inferred quartet topologies optimally. This is an important result since, to date, there have been no polynomial time algorithms with performance guarantees for quartet methods. In fact, this is the first known PTAS for inferring evolutionary trees under any paradigm. To achieve this result the natural denseness of the set Q is exploited. The second result is a new technique, called quartet cleaning, that detects and corrects errors in the set Q with performance guarantees. This result has particular significance since quartet methods are usually very sensitive to errors in the data. It is shown how quartet cleaning can dramatically increase the accuracy of quartet methods