Extremes in the Complexity of Computing Metric Distances Between Partitions

Day [3] describes an analytical model of minimum-length sequence (MLS) metrics measuring distances between partitions of a set. By selecting suitable values of model coordinates, a user may identify within the model that metric most appropriate to his classification application. Users should understand that within the model similar metrics may nevertheless exhibit extreme differences in their computational complexities. For example, the asymptotic time complexities of two MLS metrics are known to be linear in the number of objects being partitioned; yet we establish below that the computational problem for a closely related MLS metric is NP-complete.