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.
Our problem is a special case of the quadratic assignment problem (QAP): Given @math facilities, @math locations, a matrix containing the amount of flow between any pair of facilities, and a matrix containing the distances between any pair of locations. The task is to assign every facility to a location such that the cost function which is proportional to the flow between the facilities multiplied by the distances between the locations is minimized. For a survey see @cite_16 . The cost function in our problem and in the QAP are the same if we define the distances as the Manhattan distances between grid points and if we define all flows to be one. The QAP can not be approximated within any polynomial factor unless @math ; see @cite_1 . @cite_5 considered the metric version of this problem with the flow matrix being a 0 1 incidence matrix of a graph. They state some inapproximability results as well as a constant-factor approximation for the case in which the graph has vertex degree two for all but one vertex.
@cite_5 @cite_16 @cite_1
2063545374 1559577696 2028549267 2013603106
Abstract We consider the problem of placing a specified number ( p ) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This type of problem models a number of problems arising in facility location, statistical clustering, pattern recognition, and processor allocation problems in multiprocessor systems. We consider the problem under three different objectives, namely minimizing the diameter, minimizing the average distance, and minimizing the variance. We observe that, in general, the problem is NP -hard under any of the objectives. Further, even obtaining a constant factor approximation for any of the objectives is NP -hard. We present a general framework for obtaining near-optimal solutions to the compact location problems for the above measures, when the distances satisfy the triangle inequality. We show that this framework can be extended to the case when there are also node weights. Further, we investigate the complexity and approximability of more general versions of these problems, where two distance values are specified for each pair of potential sites. In these cases, the goal is to a select a specified number of facilities to minimize a function of one distance metric subject to a budget constraint on the other distance metric. We present algorithms that provide solutions which are within a small constant factor of the objective value while violating the budget constraint by only a small constant factor. In the Universal Facility Location problem we are given a set of demand points and a set of facilities. The goal is to assign the demands to facilities in such a way that the sum of service and facility costs is minimized. The service cost is proportional to the distance each unit of demand has to travel to its assigned facility, whereas the facility cost of each facility i depends on the amount of demand assigned to that facility and is given by a cost function f i (·). We present a (7.88 + e)-approximation algorithm for the Universal Facility Location problem based on local search, under the assumption that the cost functions f i are nondecreasing. The algorithm chooses local improvement steps by solving a knapsack-like subproblem using dynamic programming. This is the first constant-factor approximation algorithm for this problem. Our algorithm also slightly improves the best known approximation ratio for the capacitated facility location problem with non-uniform hard capacities. In the capacitated facility location problem with hard capacities, we are given a set of facilities, @math , and a set of clients @math in a common metric space. Each facility i has a facility opening cost f i and capacity u i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set @math and assign each client to an open facility so that at most u i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way. The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph partitioning can be formulated as a QAP. In this paper, we present some of the most important QAP formulations and classify them according to their mathematical sources. We also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms. We then give a detailed discussion of the progress made in both exact and heuristic solution methods, including those formulated according to metaheuristic strategies. Finally, we analyze the contributions brought about by the study of different approaches.