Minimum Quartet Inconsistency Is Fixed Parameter Tractable

Abstract

We study the parameterized complexity of the problem to reconstruct a binary (evolutionary) tree from a complete set of quartet topologies in the case of a limited number of errors. More precisely, we are given n taxa, exactly one topology for every subset of 4 taxa, and a positive integer k (the parameter). Then, the Minimum Quartet In- consistency (MQI) problem is the question of whether we can find an evolutionary tree inducing a set of quartet topologies that differs from the given set in only k quartet topologies. MQI is NP-complete. However, we can compute the required tree in worst case time O(4^k. n + n⁴)— the problem is fixed parameter tractable. Our experimental results show that in practice, also based on heuristic improvements proposed by us, even a much smaller exponential growth can be achieved. We extend the fixed parameter tractability result to weighted versions of the problem. In particular, our algorithm can produce all solutions that resolve at most k errors.