Consensus Tree Under the Ancestor-Descendant Distance is NP-Hard

Due to uncertainty in tumor phylogeny inference from sequencing data, many methods infer multiple, equally plausible phylogenies for the same cancer. To summarize the solution space T of tumor phylogenies, consensus tree methods seek a single best representative tree S under a specified pairwise tree distance function. One such distance function is the ancestor-descendant (AD) distance d(T,T '), which equals the size of the symmetric difference of the transitive closures of the edge sets E(T) and E(T '). Here, we show that finding a consensus tree S for tumor phylogenies T that minimizes the total AD distance n-ary sumation T is an element of Td(S,T) is NP-hard.