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.

Bad air quality due to free pollutants such as particulate matter (PM), carbon dioxide (CO 2 ), nitrogen oxides (NO x ) and volatile organic components (VOC) increases the risk of long- term health diseases. The impact of traffic-calming measures on air quality has been studied using specialized equipment at control sites or mounted on cars to monitor pollutants levels. However, this approach suffers from a large number of variables on the experiments such as vehicles types, number of monitored vehicles, driver’s behavior, traffic density, time of the day, elapsed monitoring time, road conditions and weather. In this work, we use a cellular automata and an instantaneous traffic emissions model to capture the effect of speed humps on traffic flow and on the generation of CO 2 , NO x , VOC and PM pollutants. This approach allows us to study and characterize the effect of many speed humps on a single lane. We found that speed humps significantly promote the generation of pollutants when the number of vehicles on a lane is low. Our results may provide insight into urban planning strategies to reduce the generation of traffic emissions and lower the risk of long-term health diseases.

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The capacitated vertex k-center problem receives as input a complete weighted graph and a set of capacity constraints. Its goal is to find a set of k centers and an assignment of vertices that does not violate the capacity constraints. Furthermore, the distance from the farthest vertex to its assigned center has to be minimized. The capacitated vertex k-center problem models real situations where a maximum number of clients must be assigned to centers and the travel time or distance from the clients to their assigned center has to be minimized. These centers might be hospitals, schools, police stations, among many others. The goal of this paper is to explicitly state how the capacitated vertex k-center problem and the minimum capacitated dominating set problem are related. We present an exact algorithm that consists of solving a series of integer programming formulations equivalent to the minimum capacitated dominating set problem over the bottleneck input graph. Lastly, we present an empirical evaluation of the proposed algorithm using off-the-shelf optimization software.