On the complexity of non-binary tree reconciliation with endosymbiotic gene transfer.

Reconciling a non-binary gene tree with a binary species tree can be done efficiently in the absence of horizontal gene transfers, but becomes NP-hard in the presence of gene transfers. Here, we focus on the special case of endosymbiotic gene transfers (EGT), i.e. transfers between the mitochondrial and nuclear genome of the same species. More precisely, given a multifurcated (non-binary) gene tree with leaves labeled 0 or 1 depending on whether the corresponding genes belong to the mitochondrial or nuclear genome of the corresponding species, we investigate the problem of inferring a most parsimonious Duplication, Loss and EGT (DLE) Reconciliation of any binary refinement of the tree. We present a general two-steps method: ignoring the 0-1 labeling of leaves, output a binary resolution minimizing the Duplication and Loss (DL) Reconciliation and then, for such resolution, assign a known number of 0s and 1s to the leaves in a way minimizing EGT events. While the first step corresponds to the well studied non-binary DL-Reconciliation problem, the complexity of the label assignment problem corresponding to the second step is unknown. We show that this problem is NP-complete, even when the tree is restricted to a single polytomy, and even if transfers can occur in only one direction. We present a general algorithm solving each polytomy separately, which is shown optimal for a unitary cost of operation, and a polynomial-time algorithm for solving a polytomy in the special case where genes are specific to a single genome (mitochondrial or nuclear) in all but one species. This work represents the first algorithmic study for reconciliation with endosymbiotic gene transfers in the case of a multifurcated gene tree.