Computing phylogenetic roots with bounded degrees and errors is NP-complete

In this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E), we wish to find a tree T such that (1) the degree of each internal node of T is at least 3 and at most Δ, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E and {{u,v}:u,v are leaves of T and dT(u,v)≤k}, is as small as possible, where dT(u,v) denotes the distance between u and v in tree T. We show that this problem is NP-hard for all fixed constants Δ,k≥3.Let sΔ(k) be the size of the largest clique for which an error-free tree T exists. In the course of our proof, we will determine all trees (possibly with degree 2 nodes) that approximate the (sΔ(k)-1)-clique by errors at most 2.