Abstract In a hierarchical clustering problem the task is to compute a series of mutually compatible clusterings of a finite metric space $$(P,{{\,\textrm{dist}\,}})$$ ( P , dist ) . Starting with the clustering where every point forms its own cluster, one iteratively merges two clusters until only one cluster remains. Complete linkage is a well-known and popular algorithm to compute such clusterings: in every step it merges the two clusters whose union has the smallest radius (or diameter) among all currently possible merges. We prove that the radius (or diameter) of every k-clustering computed by complete linkage is at most by factor O(k) (or $$O(k^{\ln (3)/\ln (2)})=O(k^{1{.}59})$$ O ( k ln ( 3 ) / ln ( 2 ) ) = O ( k 1.59 ) ) worse than an optimal k-clustering minimizing the radius (or diameter). Furthermore we give a negative answer to the question proposed by Dasgupta and Long (J Comput Syst Sci 70(4):555–569, 2005. https://doi.org/10.1016/j.jcss.2004.10.006 ), who show a lower bound of $$\Omega (\log (k))$$ Ω ( log ( k ) ) and ask if the approximation guarantee is in fact $$\Theta (\log (k))$$ Θ ( log ( k ) ) . We present instances where complete linkage performs poorly in the sense that the k-clustering computed by complete linkage is off by a factor of $$\Omega (k)$$ Ω ( k ) from an optimal solution for radius and diameter. We conclude that in general metric spaces complete linkage does not perform asymptotically better than single linkage, merging the two clusters with smallest inter-cluster distance, for which we prove an approximation guarantee of O(k).

We study Ward's method for the hierarchical k-means problem. This popular greedy heuristic is based on the complete linkage paradigm: Starting with all data points as singleton clusters, it successively merges two clusters to form a clustering with one cluster less. The pair of clusters is chosen to (locally) minimize the k-means cost of the clustering in the next step. Complete linkage algorithms are very popular for hierarchical clustering problems, yet their theoretical properties have been studied relatively little. For the Euclidean k-center problem, Ackermann et al. [1] show that the k-clustering in the hierarchy computed by complete linkage has a worst-case approximation ratio of Θ(log k). If the data lies in ℝd for constant dimension d, the guarantee improves to O(1) [23], but the O-notation hides a linear dependence on d. Complete linkage for k-median or k-means has not been analyzed so far. In this paper, we show that Ward's method computes a 2-approximation with respect to the k-means objective function if the optimal k-clustering is well separated. If additionally the optimal clustering also satisfies a balance condition, then Ward's method fully recovers the optimum solution. These results hold in arbitrary dimension. We accompany our positive results with a lower bound of Ω((3/2)d) for data sets in ℝd that holds if no separation is guaranteed, and with lower bounds when the guaranteed separation is not sufficiently strong. Finally, we show that Ward produces an O(1)-approximative clustering for one-dimensional data sets.

We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number n of variables, the number m of constraints, and 1/\delta, where \delta is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. n, m, and 1/delta) between two given vertices of a polyhedron [4]. Our result also complements a recent result due to Eisenbrand and Vempala [6] who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables n and 1/\delta, but independent of the number m of constraints. Even though the running time of our algorithm depends on m, it is significantly faster for the important special case of totally unimodular linear programs, for which 1/delta\le n and which have only O(n^2) constraints.