Abstract. We explore the use of instance and cluster-level constraints with agglomerative hierarchical clustering. Though previous work has illustrated the benefits of using constraints for non-hierarchical clustering, their application to hierarchical clustering is not straight-forward for two primary reasons. First, some constraint combinations make the feasibility problem (Does there exist a single feasible solution?) NP-complete. Second, some constraint combinations when used with traditional agglomerative algorithms can cause the dendrogram to stop prematurely in a dead-end solution even though there exist other feasible solutions with a significantly smaller number of clusters. When constraints lead to efficiently solvable feasibility problems and standard agglomerative algorithms do not give rise to dead-end solutions, we empirically illustrate the benefits of using constraints to improve cluster purity and average distortion. Furthermore, we introduce the new γ constraint and use it in conjunction with the triangle inequality to considerably improve the efficiency of agglomerative clustering. 1

Many clustering applications use the computationally efficient non-hierarchical clustering techniques such as k-means. However, less efficient hierarchical clustering is desirable as by creating a dendrogram the user can choose an appropriate value of k (the number of clusters) and in some domains cluster hierarchies (i.e. clusters within other clusters) naturally exist. In many situations apriori constraints/information are available such as in the form of a small amount of labeled data. In this paper we explore using constraints to improve the e#- ciency of agglomerative clustering algorithms. We show that just finding feasible (satisfying all constraints) solutions for some constraint combinations is NP-complete and should be avoided. For a given set of constraints we derive upper (kmax ) and lower bounds (kmin ) on the value of k where feasible solutions exist. This allows a restricted dendrogram to be created but its creation is not straight-forward. For some combinations of constraints, starting with a feasible clustering solution (k = r) and joining the two closest clusters results in a "dead-end" feasible solution which cannot be further refined to create a feasible solution with r - 1 clusters even though kmin - 1 kmax . For such situations we introduce constraint driven hierarchical clustering algorithms that will create a complete dendrogram. When traditional algorithms can be used, we illustrate the use of the triangle inequality and a newly defined # constraint to further improve performance and use the Markov inequality to bound the expected performance improvement. Preliminary results indicate that using constraints can improve the dendrogram quality.

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