An n-town, n@?N, is a group of n buildings, each occupying a distinct position on a 2-dimensional integer grid. If we measure the distance between two buildings along the axis-parallel street grid, then an n-town has optimal shape if the sum of all pairwise Manhattan distances is minimized. This problem has been studied for cities, i.e., the limiting case of very large n. For cities, it is known that the optimal shape can be described by a differential equation, for which no closed-form solution is known. We show that optimal n-towns can be computed in O(n^7^.^5) time. This is also practically useful, as it allows us to compute optimal solutions up to n=80.
@cite_10: The metric 2-clustering problem is defined as follows: given a metric (or weighted graph) (X,d), partition X into two sets S(1) and S(2) in order to minimize the value of spl Sigma sub i spl Sigma sub u,v spl sub S(i) d(u,v). In this paper, we show an approximation scheme for this problem.
@cite_3: The min-sum k-clustering problem in a metric space is to find a partition of the space into k clusters as to minimize the total sum of distances between pairs of points assigned to the same cluster. We give the first polynomial time non-trivial approximation algorithm for this problem. The algorithm provides an @math approximation to the min-sum k -clustering problem in general metric spaces, with running time @math . The result is based on embedding of metric spaces into hierarchically separated trees. We also provide a bicriteria approximation result that provides a constant approximation factor solution with only a constant factor increase in the number of clusters. This result is obtained by modifying and drawing ideas from recently developed primal dual approximation algorithms for facility location.
@cite_1: For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.
@cite_20: 
Another related problem is called or . The goal is to separate a graph into @math clusters to minimize the sum of pairwise distances between nodes in the same cluster. For general graphs, @cite_1 show that this problem is NP-hard to approximate within any constant factor for @math . In a metric space the problem is easier to approximate: @cite_20 give a @math -approximation, @cite_10 gives a PTAS for @math , and @cite_3 give an @math -approximation for general @math .