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_13: This paper concerns the problem of computing the densest k-vertex subgraph of a given graph, namely, the subgraph with the most edges, or with the highest edges-to-vertices ratio. A sequence of approximation algorithms is developed for the problem, with each step yielding a better ratio at the cost of a more complicated solution. The approximation ratio of our final algorithm is O spl tilde (n sup 0.3885 ). We also present a method for converting an approximation algorithm for an unweighted graph problem (from a specific class of maximization problems) into one for the corresponding weighted problem, and apply it to the densest subgraph problem. >
@cite_7: Given an n-vertex graph with nonnegative edge weights and a positive integer k?n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1 2+n 2k)2?O(n?1 3)?R?(1 2+n 2k)2+O(1 n) for k in the range n 3?k?n and 2(n k?1)?O(1 k)?R?2(n k?1)+O(n k2) for k
@cite_17: We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimumk-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree?(n), although our algorithms solve most of these problems so long as the average degree is?(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea ofexhaustive sampling:picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certainsmoothinteger programs, where the objective function and the constraints are “dense” polynomials of constant degree.
The reverse version of the discrete problem, where the goal is to maximize the average distance between points, has also been studied: In the maximization version, called the , the objective is to pick @math points from a set of size @math so that the pairwise distance is maximized. When the edge weights need not obey the triangle inequality, @cite_13 give an @math -approximation. @cite_7 improve this guarantee to a constant factor in the special case when @math and @cite_17 give a PTAS when @math and @math .