Geometric medians in reconciliation spaces of phylogenetic trees

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.