We investigate the simultaneous existence of two centers (bi-center) for two families of planar Z2-equivariant differential systems. First we present necessary and sufficient conditions for the existence of an isochronous bi-center for a planar Z2-equivariant cubic system having two centers at the points (−1, 0) and (1, 0), completing the study done by Liu and Li (2011). Next, we give condition...

In the k-center problem, the input is a bound k and n points with the distance between every two of them, such that the distances obey the triangle inequality. The goal is to choose a set of k points to serve as centers, so that the maximum distance from the centers C to any point is as small as possible. This fundamental facility location problem is NP-hard. The symmetric case is well-understo...

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We propose a new hybrid symbolic-numerical approach to the center-focus problem. The method allowed us to obtain center conditions for a three-dimensional system of differential equations, which was previously not possible using traditional, purely symbolic computational techniques.

The p-center problem is one of the location problems that have been studied in operations research and computational geometry. This paper describes a compatible discrete space version of the heuristic Voronoi diagram algorithm. Since the algorithm gets stuck in local optimums in some cases, we apply a number of changes in the body of the algorithm with regard to the geometry of the problem, in ...

The basic K-center problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. This problem is known to be NP-hard, and several optimal approximation algorithms that achieve a factor of 2 have been developed for it. We ...

Given a convex polygon P with n vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover P . We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius r satisfying r ≤ 2ropt using O(1) space, where ropt is the optimum solution. Nex...

We give a review of existing methods for solving the absolute and vertex restricted p-center problems on networks and propose a new integer programming formulation, a tightened version of this formulation and a new method based on successive restrictions of the new formulation. A specialization of the new method with two-element restrictions obtains the optimal p-center solution by solving a se...

In an inverse network 1-center location problem the parameters of a given network, like edge lengths or vertex weights, have to be modified at minimum total cost such that a prespecified vertex s becomes a 1-center of the network. In this paper, we consider the inverse 1-center location problem on an unweighted tree network T with n + 1 vertices in which we are allowed to increase or reduce the...

The classical Center-Focus problem posed by H. Poincaré in 1880’s asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point (which is called a center). In this paper we describe a new general approach to the Center Problem.