A fixed-parameter algorithm for minimum quartet inconsistency - Gramm, Jens; Niedermeier, Rolf (Journal of computer and system sciences, 2003)
Given n taxa, exactly one topology for every subset of four taxa, and a positive integer k (the parameter), the M inimum Q uartet I nconsistency (MQI) problem is the question whether we can find an evolutionary tree inducing a set of quartet topologies that differs from the given set in only k quartet topologies. The more general problem where we are not necessarily given a topology for every subset of four taxa appears to be fixed-parameter intractable. For MQI, however, which is also NP-complete, we can compute the required tree in time O(4 k n+ n 4). This means that the problem is fixed-parameter tractable and that in the case of a small number k of “errors” the tree reconstruction can be done efficiently. In particular, for minimal k, our algorithm can produce all solutions that resolve k errors. Additionally, we discuss significant heuristic improvements. Experiments underline the practical relevance of our solutions.