Rectilinear static and dynamic discrete 2-center problems

Computational Geometry

We consider the planar Euclidean two-center problem in which given n points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic O(n log n)-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic O(n log n)-time algorithm for the case that the input points are in convex position. Both results improve the previous best O(n log n log log n) bound on the problems.

2015 IEEE International Conference on Service Operations And Logistics, And Informatics (SOLI), 2015

The facility location problem is a well-known challenge in logistics that is proven to be NP-hard. In this paper we specifically simulate the geographical placement of facilities to provide adequate service to customers. Determining reasonable center locations is an important challenge for a management since it directly effects future service costs. Generally, the objective is to place the central nodes such that all customers have convenient access to them. We analyze the problem and compare different placement strategies and evaluate the number of required centers. We use several existing approaches and propose a new heuristic for the problem. For our experiments we consider various scenarios and employ simulation to evaluate the performance of the optimization algorithms. Our new optimization approach shows a significant improvement. The presented results are generally applicable to many domains, e.g., the placement of military bases, the planning of content delivery networks, or the placement of warehouses.

International Conference on Computing: Theory and Applications, 2007

We consider the problem of nding two parallel rectangles, in arbitrary orientation, covering a given set of n points in a plane, such that the area of the larger rectangle is minimized. We give a simple algorithm that solves the problem in O(n3) time using O(n2) space. Without altering the complexity, the algorithm can be modied to solve another optimization

Open Computer Science, 2020

A two-stage continuous-discrete optimal partitioning-allocation problem is studied, and a method and an algorithm for its solving are proposed. This problem is a generalization of a classical transportation problem to the case when coordinates of the production points (collection, storage, processing) of homogeneous products are continuously allocated in the given domain and the production volumes at these points are unknown. These coordinates are found as a solution of the corresponding continuous optimal set-partitioning problem in a finitedimensional Euclidean space with the placement (finding coordinates) of these subsets' centers. Also, this problem generalizes discrete two-stage production-transportation problems to the case of continuously allocated consumers. The method and algorithm are illustrated by solving two model problems.

Naval Research Logistics Quarterly, 1981

The location-allocation problem for existing facilities uniformly distributed over rectangular regions is treated for the case where the rectilinear norm is used. The new facilities are to be located such that the expected total weighted distance is minimized. Properties of the problem are discussed. A branch and bound algorithm is developed for the exact solution of the problem. Computational results are given for different sized problems.

Naval Research Logistics, 2003

When solving location problems in practice it is quite common to aggregate demand points into centroids. Solving a location problem with aggregated demand data is computationally easier, but the aggregation process introduces error. We develop theory and algorithms for certain types of centroid aggregations for rectilinear 1-median problems. The objective is to construct an aggregation that minimizes the maximum aggregation error. We focus on rowcolumn aggregations, and make use of aggregation results for 1-median problems on the line to do aggregation for 1-median problems in the plane. The aggregations developed for the 1-median problem are then used to construct approximate n-median problems. We test the theory computationally on n-median problems (n Ն 1) using both randomly generated, as well as real, data. Every error measure we consider can be well approximated by some power function in the number of aggregate demand points. Each such function exhibits decreasing returns to scale.

Lecture Notes in Computer Science, 2012

We study a class of geometric optimization problems closely related to the 2-center problem: Given a set S of n pairs of points, assign to each point a color ("red" or "blue") so that each pair's points are assigned different colors and a function of the radii of the minimum enclosing balls of the red points and the blue points, respectively, is optimized. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii. For each case, minmax and minsum, we consider distances measured in the L2 and in the L∞ metrics. Our problems are motivated by a facility location problem in transportation system design, in which we are given origin/destination pairs of points for desired travel, and our goal is to locate an optimal road/flight segment in order to minimize the travel to/from the endpoints of the segment.

Discrete & Computational Geometry, 1996

Let P be a set of n points in the plane and let e be a segment of fixed length. The segment-center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(nl+~), for any e > 0, improving the previous solution of Agarwal et al. [3] by nearly a factor of O (n).

Theoretical Computer Science, 2016

In typical graph minimization problems, we consider a graph G with fixed weights on the edges of G. The goal is then to find an optimal vertex or set of vertices with respect to some objective function, for example. We introduce a new framework for graph minimization problems, where the weights on the graph edges are not fixed, but rather must be assigned, and the weight is inversely proportional to the cost paid. The goal is to find a valid assignment for which the resulting weighted graph optimizes the objective function. We present algorithms for finding the optimal budget allocation for the center point problem and for the median point problem on trees. Our algorithms run in linear time, both for the case where a candidate vertex is given as part of the input, and for the case where finding a vertex that optimizes the solution is part of the problem. We also present a hardness result for the center point problem on complete metric graphs, followed by an O (log 2 (n)) approximation algorithm in this setting.

2002

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height.