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We present a new multi-layer peeling technique to cluster points in a metricspace. A well-known non-parametric objective is to embed the metric space intoa simpler structured metric space such as a line (i.e., Linear Arrangement) ora binary tree (i.e., Hierarchical Clustering). Points which are close in themetric space should be mapped to close points/leaves in the line/tree;similarly, points which are far in the metric space should be far in the lineor on the tree. In particular we consider the Maximum Linear Arrangementproblem \cite{Approximation_algorithms_for_maximum_linear_arrangement} and theMaximum Hierarchical Clustering problem\cite{Hierarchical_Clustering:_Objective_Functions_and_Algorithms} applied tometrics.  We design approximation schemes ($1 - \epsilon$ approximation for anyconstant $\epsilon > 0$) for these objectives. In particular this shows that byconsidering metrics one may significantly improve former approximations ($0.5$for Max Linear Arrangement and $0.74$ for Max Hierarchical Clustering). Ourmain technique, which is called multi-layer peeling, consists of recursivelypeeling off points which are far from the "core" of the metric space. Therecursion ends once the core becomes a sufficiently densely weighted metricspace (i.e. the average distance is at least a constant times the diameter) oronce it becomes negligible with respect to its inner contribution to theobjective. Interestingly, the algorithm in the Linear Arrangement case is muchmore involved than that in the Hierarchical Clustering case, and uses asignificantly more delicate peeling.

Multi Layer Peeling for Linear Arrangement and Hierarchical Clustering