Approximating minimum quartet inconsistency (abstract)

A fundamental problem in computational biology which has been widely studied in the last decades is the reconstruction of evolutionary trees from biological data. Unfortunately, almost all its known formulations are NPhard. The compelling need for having efficient computational tools to solve this biological problem has brought a lot of attention to the analysis of the quartet paradigm for inferring evolutionary trees [1, 2, 8]. Given a quartet of taxa {a, b, c, d}, there are 3 possible degree-3 trees connecting the taxa as terminals. Each such tree is called a quartet topology. The quartet methods proceed by first estimating the topology of each quartet of taxa and then recombining the inferred quartet topologies into an evolutionary tree. A major difficulty in this approach derives from the fact that quartet topology inference methods often make mistakes, and thus may result in a set Q of quartet topologies that is inconsistent with any evolutionary tree. Therefore, the problem of recombining the quartet topologies of Q to form an estimate of the correct evolutionary tree is naturally formulated as an optimization problem that looks for a tree T maximizing the number of consistent quartets (i.e. |Q ∩ QT |), or equivalently minimizing the number of inconsistent quartets (i.e. |Q − QT |), where QT denotes the unique set of quartet topologies induced by T . The above (complementary) problems are referred to as Maximum Quartet Consistency (MQC) and Minimum Quartet Inconsistency (MQI) problems. In a recent paper [4], it has been shown that MQC is NP-hard, but it admits a PTAS, using the technique of smooth integer polynomial programming and exploiting the natural denseness of the set Q (i.e. |Q| = ( n 4 )