Consensus properties and their large-scale applications for the gene duplication problem.

Solving the gene duplication problem is a classical approach for species tree inference from gene trees that are confounded by gene duplications. This problem takes a collection of gene trees and seeks a species tree that implies the minimum number of gene duplications. Wilkinson et al. posed the conjecture that the gene duplication problem satisfies the desirable Pareto property for clusters. That is, for every instance of the problem, all clusters that are commonly present in the input gene trees of this instance, called strict consensus, will also be found in every solution to this instance. We prove that this conjecture does not generally hold. Despite this negative result we show that the gene duplication problem satisfies a weaker version of the Pareto property where the strict consensus is found in at least one solution (rather than all solutions). This weaker property contributes to our design of an efficient scalable algorithm for the gene duplication problem. We demonstrate the performance of our algorithm in analyzing large-scale empirical datasets. Finally, we utilize the algorithm to evaluate the accuracy of standard heuristics for the gene duplication problem using simulated datasets.