Reconstructing a SuperGeneTree minimizing reconciliation

Combining a set of trees on partial datasets into a single tree is a classical method for inferring large phylogenetic trees. Ideally, the combined tree should display each input partial tree, which is only possible if input trees do not contain contradictory phylogenetic information. The simplest version of the supertree problem is thus to state whether a set of trees is compatible, and if so, construct a tree displaying them all. Classically, supertree methods have been applied to the reconstruction of species trees. Here we rather consider reconstructing a super gene tree in light of a known species tree S. We define the supergenetree problem as finding, among all supertrees displaying a set of input gene trees, one supertree minimizing a reconciliation distance with S. We first show how classical exact methods to the supertree problem can be extended to the supergenetree problem. As all these methods are highly exponential, we also exhibit a natural greedy heuristic for the duplication cost, based on minimizing the set of duplications preceding the first speciation event. We then show that both the supergenetree problem and its restriction to minimizing duplications preceding the first speciation are NP-hard to approximate within a n(1-ϵ )factor, for any 0 < ϵ < 1. Finally, we show that a restriction of this problem to uniquely labeled speciation gene trees, which is relevant to many biological applications, is also NP-hard. Therefore, we introduce new avenues in the field of supertrees, and set the theoretical basis for the exploration of various algorithmic aspects of the problems.