On the Midpath Tree Conjecture A Counter Example

Introduction Clustering data based on pairwise distances is a fundamental problem Weighted trees can be used to represent hierarchical clusters Multidi mensional Scaling MDS in some formulations is the problem of clustering data based on preserving their relative rather than absolute distances We call this an ordinal clustering A particular instance of this problem has been con sidered in the algorithmic computational biology com munity KW KHM Given a total or partial order on the pairwise distances between points give a weighted tree on those points so that the pairwise pathlengths between points in the tree satis es this order This is therefore simply the MDS problem for hierarchical clustering and we will refer to it as the HMDS prob lem It has been conjectured Kan that there is a polyno mial time algorithm to solve the HMDS problem both while preserving the total order of the points as well as while relaxing the order to certain types of partial orders This algorithm is called the Midpath Tree Al gorithm While it clearly runs in polynomial time it was not known to always produce the correct output for HMDS In this short paper we show that it does not in fact always produce the correct output Previous work includes an algorithm described by Kearney et al in KHM to construct an unweighted tree if it exists which realizes the total order on pair wise distances Clearly in many cases an unweighted tree may not exist for an order while a weighted tree does exist so solving the unweighted case sheds little light on the weighted HMDS problem Kannan and Warnow KW solved a similar problem to realize cer tain types of partial orders and posed the weighted case as an open problem They speci cally worked with a partial order constructed from triplets of data points which we call a triangle order A triangle order is a partial order on distances so that that distances within each triplet of points are totally ordered Before presenting the Midpath Tree Algorithm and its counter example we present some preliminary ob servations