A Note on the Fixed Parameter Tractability of the Gene-Duplication Problem.

The NP-hard gene-duplication problem takes as input a collection of gene trees and seeks a species tree that requires the fewest number of gene duplications to reconcile the input gene trees. An oft-cited, decade-old result by Stege states that the gene-duplication problem is fixed parameter tractable when parameterized by the number of gene duplications necessary for the reconciliation. Here, we uncover an error in this fixed parameter algorithm and show that this error cannot be corrected without sacrificing the fixed parameter tractability of the algorithm. Furthermore, we show a link between the gene-duplication problem and the minimum rooted triplets inconsistency problem which implies that the gene-duplication problem is 1) W[2]-hard when parameterized by the number of gene duplications necessary for the reconciliation and 2) hard to approximate to better than a logarithmic factor.