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The vertex k-center problem is a classical NP-Hard optimization problem with application to Facility Location and Clustering among others. This problem consists in finding a subset C ⊆ V of an input graph G = (V , E), such that the distance from the farthest vertex in V to its nearest center in C is minimized, where |C| ≤ k, with k ∈ Z + as part of the input. Many heuristics, metaheuristics, approximation algorithms, and exact algorithms have been developed for this problem. This paper presents an analytical study and experimental evaluation of the most representative approximation algorithms for the vertex k-center problem. For each of the algorithms under consideration and using a common notation, we present proofs of their corresponding approximation guarantees as well as examples of tight instances of such approximation bounds, including a novel tight example for a 3-approximation algorithm. Lastly, we present the results of extensive experiments performed over de facto benchmark data sets for the problem which includes instances of up to 71009 vertices. INDEX TERMS Approximation algorithms, k-center problem, polynomial time heuristics. I. INTRODUCTION Perhaps one of the first center selection problems for which there is historical register is the following: ''given three points in the plane, find a fourth point such that the sum of its distances to the three points is minimized'' [1]. Given its simplicity, it is hard to establish who first stated this problem. However, this problem is usually associated to Pierre de Fermat, who asked this question around 1636, and its first registered solution is associated to Evangelista Torricelli [1]. An extension of this problem is known as the Weber's problem, where the points have an associated cost and the goal is to locate not 1 but k centers [1]. By adding new properties and restrictions to a basic k-center problem, the collection of k-center problems have become larger over the years. One of the basic center selection problems that more directly gave rise to many other center problems is known The associate editor coordinating the review of this manuscript and approving it for publication was Diego Oliva.

Laboratoire de Magn etisme et de Physique des Hautes Energies (URAC 12), D epartement de physique B.P. 1014, Facult e des sciences Universit e Mohammed V Agdal, Rabat, MoroccoIn this paper, using Nagel-Schreckenberg model we study the on-ramp system under the expanded open boundary condition. The phase diagram of the two-lane on-ramp system is computed. It is found that the expanded left boundary insertion strategy enhances the°ow in the on-ramp lane. Furthermore, we have studied the probability of the occurrence of car accidents. We distinguish two types of car accidents: the accident at the on-ramp site ðP rc Þ and the rearend accident in the main road ðP ac Þ. It is shown that car accidents at the on-ramp site are more likely to occur when tra±c is free on road A. However, the rear-end accidents begin to occur above a critical injecting rate c1 . The in°uence of the on-ramp length ðL B Þ and position ðx C 0 Þ on the car accidents probabilities is studied. We found that large L B or x C 0 causes an important decrease of the probability P rc . However, only large x C 0 provokes an increase of the probability P ac . The e®ect of the stochastic randomization is also computed.

In this paper, using the Nagel-Schreckenberg model, we numerically investigate the probability P ac of entering/circulating car accidents to occur at single-lane roundabout under the expanded open boundary. The roundabout consists of N on-ramps (respectively, o®-ramps). The boundary is controlled by the injecting rates 1 ; 2 and the extracting rate . The simulation results show that, depending on the injecting rates, the car accidents are more likely to happen when the capacity of the rotary is set to its maximum. Moreover, we found that the large values of rotary size L and the probability of preferential P exit are reliable to improve safety and reduce accidents. However, the usage of indicator, the increase of and/or N provokes an increase of car accident probability.

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The correlation between the velocity of two successive vehicles as a function of time headway is studied in the one-dimensional cellular automata (CA) NaSch model within parallel dynamic update. It is found that a strong correlation occurs in short time headway. The behavior of the correlation velocity as a function of the car density in different traffic states is also investigated. Moreover, our study is also extended to a more complicated situation where the two vehicles are separated by a number n of other vehicles. It is shown that the velocity correlation coefficient depends strongly on the number n of vehicles in between and on their positions.