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, a center. In this paper we describe a new general approach to the Center Problem.

The set of vertices that maximize (minimize) the remoteness is the antimedian (median) set of the profile. It is proved that for an arbitrary graph G and S ⊆ V (G) it can be decided in polynomial time whether S is the antimedian set of some profile. Graphs in which every antimedian set is connected are also considered.

A problem that is constantly cropping up in designing even the simplest algorithm or a program is dealing with ±1 bug, or one-off bug, when we calculate positions within an array, very noticeably while splitting it in half. This bug is often found in buffer overflow type of bugs. While designing one complicated algorithm, we needed various ways of splitting an array, and we found the lack of ge...

Locating facilities on anywhere on a network is known as the absolute p-center location problem and it is proven to be NP-hard problem. Most of the recent approaches solve largescale vertex p-center location problem in which facilities can be located only on the nodes of the network. However, rarely algorithms are developed to solve large-scale absolute p-center problem. Particle swarm optimiza...

In this paper we deal with the vertex k-center problem, a problem which is a part of the discrete location theory. Informally, given a set of cities, with intercity distances specified, one has to pick k cities and build warehouses in them so as to minimize the maximum distance of any city from its closest warehouse. We examine several approximation algorithms that achieve approximation factor ...

We study a generalization of the p{Center Problem, which we call the {Neighbor p{Center Problem (p{Center ()). Given a complete edge{ weighted network, the goal is to minimize the maximum distance of a client to its nearest neighbors in the set of p centers. We show that in general nding a O(2 poly(jV j)){approximation for p{ Center () is NP{hard, where jV j denotes the number of nodes in the n...

Given a metric graph G = (V,E,w) and a positive integer k, the Single Allocation k-Hub Center problem is to find a spanning subgraph H of G such that (i) C ⊂ V is a clique of size k in H; (ii) V \ C forms an independent set in H; (iii) each v ∈ V \C is adjacent to exactly one vertex in C; and (iv) the diameter D(H) is minimized. The vertices selected in C are called hubs and the rest of vertice...