Hierarchical Clustering via Spreading Metrics
We study the cost function for hierarchical clusterings introduced by [16] where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown in [16] that a top-down algorithm returns a hierarchical clustering of cost at most O (α n log n) times the cost of the optimal hierarchical clustering, where α n is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top-down algorithm returns a hierarchical clustering of cost at most O log 3/2 n times the cost of the optimal solution. We improve this by giving an O(log n)-approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an O(log n)-approximate hierarchical clustering for a generalization of this cost function also studied in [16]. We also give constant factor inapproximability results for this problem.
I enjoyed reading this paper which shows that the hierarchical clustering problem [6] can be approximated within a multiplicative O(log n) factor. This improves the O(\alpha log n) approximation of [6] where \alpha is an approximation factor of a subroutine used to solve the “Sparsest Cut” problem. In order to avoid losing the approximation factor of the “Sparsest Cut” problem the authors develop a new approach that is based on a careful LP rounding algorithm. In a nutshell, an ILP formulation is used to give a characterization of ultrametrics associated with the problem. The key observation here is that the ultrametrics associated with the hierarchical clustering problem have to satisfy a non-triviality condition (Def 1 + Lem 5). This condition is then captured by the ILP together with the ultrametric condition (for the latter the techniques of [8] are used). Then an LP-relaxation based algorithm with iterative rounding is given that achieves the desired approximation using Leighton-Rao “sphere growing” technique. It is quite remarkable that a fairly complex LP-formulation used in this work does in fact admit such a rounding approach.   I also very much enjoyed the overall comprehensiveness of the approach taken in this paper as it includes experimental evaluation on datasets from the UCI database + negative results that show superpolynomial lower bounds on the LP/SDP relaxations of the problem. 
I enjoyed reading this paper which shows that the hierarchical clustering problem [6] can be approximated within a multiplicative O(log n) factor. This improves the O(\alpha log n) approximation of [6] where \alpha is an approximation factor of a subroutine used to solve the “Sparsest Cut” problem. In order to avoid losing the approximation factor of the “Sparsest Cut” problem the authors develop a new approach that is based on a careful LP rounding algorithm. In a nutshell, an ILP formulation is used to give a characterization of ultrametrics associated with the problem. The key observation here is that the ultrametrics associated with the hierarchical clustering problem have to satisfy a non-triviality condition (Def 1 + Lem 5). This condition is then captured by the ILP together with the ultrametric condition (for the latter the techniques of [8] are used). Then an LP-relaxation based algorithm with iterative rounding is given that achieves the desired approximation using Leighton-Rao “sphere growing” technique. It is quite remarkable that a fairly complex LP-formulation used in this work does in fact admit such a rounding approach.   I also very much enjoyed the overall comprehensiveness of the approach taken in this paper as it includes experimental evaluation on datasets from the UCI database + negative results that show superpolynomial lower bounds on the LP/SDP relaxations of the problem.