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_9: The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG). We consider the question of obtaining near-optimal solutions. for MAX-MIN, we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP. When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. for MAX-AVG, we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-ti...
@cite_14: We consider a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.
@cite_6: We describe approximation algorithms with bounded performance guarantees for the following problem: A graph is given with edge weights satisfying the triangle inequality, together with two numbers k and p. Find k disjoint subsets of p vertices each, so that the total weight of edges within subsets is maximized
When the edge weights obey the triangle inequality, @cite_9 give a @math -approximation that runs in @math time and @cite_6 give a @math -approximation that runs in @math time. For points in the plane and Euclidean distances, @cite_9 give an approximation with performance bound arbitrarily close to @math . For Manhattan distances, @cite_14 give an optimal algorithm for fixed @math and a PTAS for general @math . Moreover, they provide a @math -Approximation for Euclidean distances.