On the Complexity of Ordinal Clustering

Given a set of pairwise distances on a set of n points con structing an edge weighted tree whose leaves are these n points such that the tree distances would mimic the original distances under some criteria is a fundamental problem For example this problem is sometimes called the heirarchical clustering problem One distance preservation criterion is to preserve the total order of pair wise distances We show that the problem of nding a weighted tree if it exists which would preserve the total order on pairwise distances is NP hard A partial order on pairwise distances between points in which orders all distances that share an end point so that each point has a view of the other points that is consistent with the original distances is called a triangle order since it is equivalent to an order where this distances in each triangle are ordered This order has been studied in biological settings We also show the NP hardness of the problem of nding the weighted tree which would preserve a triangle order