  Hierarchical Clustering (HC) is a widely studied problem in exploratory data
analysis, usually tackled by simple agglomerative procedures like
average-linkage, single-linkage or complete-linkage. In this paper we focus on
two objectives, introduced recently to give insight into the performance of
average-linkage clustering: a similarity based HC objective proposed by
[Moseley and Wang, 2017] and a dissimilarity based HC objective proposed by
[Cohen-Addad et al., 2018]. In both cases, we present tight counterexamples
showing that average-linkage cannot obtain better than 1/3 and 2/3
approximations respectively (in the worst-case), settling an open question
raised in [Moseley and Wang, 2017]. This matches the approximation ratio of a
random solution, raising a natural question: can we beat average-linkage for
these objectives? We answer this in the affirmative, giving two new algorithms
based on semidefinite programming with provably better guarantees.
