Geometric medians in reconciliation spaces of phylogenetic trees

In evolutionary biology, it is common to study how various entities evolve together, for example, how parasites coevolve with their host, or genes with their species. Coevolution is commonly modelled by considering certain maps or reconciliations from one evolutionary tree P to another H, all of which induce the same map ϕ between the leaf-sets of P and H (corresponding to present-day associations). Recently, there has been much interest in studying spaces of reconciliations, which arise by defining some metric d on the set R(P,H,ϕ) of all possible reconciliations between P and H. In this paper, we study the following question: How do we compute a geometric median for a given subset Ψ of R(P,H,ϕ) relative to d, i.e. an element ψmed∈R(P,H,ϕ) such that ∑ψ′∈Ψd(ψmed,ψ′)≤∑ψ′∈Ψd(ψ,ψ′) holds for all ψ∈R(P,H,ϕ)? For a model where so-called host-switches or transfers are not allowed, and for a commonly used metric d called the edit-distance, we show that it is possible to compute a geometric median for a set Ψ in R(P,H,ϕ) in polynomial time. We expect that this result could open up new directions for computing a consensus for a set of reconciliations.