When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem

García-Díaz, Jesús; Sanchez-Hernandez, Jairo Javier; Menchaca-Mendez, Rolando; Menchaca-Mendez, Rolando

doi:10.1007/s10732-017-9345-x

Cited by 12 publications

(8 citation statements)

References 16 publications

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“…This way, the uncapacitated vertex k-center problem (best known as the vertex k-center problem) aims at minimizing the distance from the farthest vertex to its nearest center. Many approximation algorithms [13][14][15][16][17][18], heuristics [19,20], metaheuristics [21][22][23][24][25], and exact algorithms have been proposed for this problem [26][27][28][29][30][31][32].…”

Section: The Capacitated Vertex K-center Problemmentioning

confidence: 99%

“…Let C be the set of selected centers. At (18), the matrix defined by all the a ij values represents the adjacency matrix of the bottleneck graph G r(C * ) = (V, E r(C * ) ), which results from removing all the edges with weight greater than r(C * ) from the original input graph G = (V, E), where r(C * ) is the size of the optimal solution C * for the uncapacitated vertex k-center problem over the input graph G = (V, E). Since simple graphs do not have loops, we set a ij to 0 for i = j.…”

Section: Definitionmentioning

confidence: 99%

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Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem

et al. 2020

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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.

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Section: The Capacitated Vertex K-center Problemmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem

et al. 2020

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“…Among the polynomial heuristic algorithms are the Gr greedy pure algorithm [27], [28], the Scr algorithm [28], and the CDSh algorithm [34]; the last two being considered as the polynomial time algorithms for the vertex k-center problem with best empirical performance [28], [29], [34]. Among the exact algorithms are those proposed by Daskin [35], Ilhan et al [36], Elloumi et al [37], Al-Khedhairi and Salhi [38], Chen and Chen [39], Calik and Tansel [40], and Contardo et al [23].…”

Section: The Vertex K-center Problemmentioning

confidence: 99%

“…Although these methods give no guarantee either the quality of the solutions they find nor the execution time, all of them perform better than the 2-approximated Sh, Gon, and HS algorithms on most benchmark data sets reported in the literature. Because of this, we also describe a fourth approximation algorithm, the CDS algorithm [34], which despite having a sub-optimal approximation factor of 3, significantly outperforms the known 2approximation algorithms over the de facto benchmark data sets from the literature. To some extent, the CDS algorithm achieve a balance between the advantages of the approximation algorithms and the other algorithmic approaches, because it runs in polynomial time, it guarantees that the quality of the solution is not arbitrary, and the empirical results show that the generated solutions are not only theoretically near optimal, but also competitive in practical terms.…”

Section: The Vertex K-center Problemmentioning

confidence: 99%

Approximation Algorithms for the Vertex K-Center Problem: Survey and Experimental Evaluation

García-Díaz

Menchaca-Mendez²,

Menchaca-Mendez³

et al. 2019

IEEE Access

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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.

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“…Drezner and Drezner [33] presented a heuristic-based greedy random adaptive algorithm for solving the sequential location problem of two facilities. Garcia-Diaz et al [34] proposed a heuristic algorithm with algorithmic complexity for solving the p-center facility location problem. Garcia-Diaz et al [35] used a structure-driven randomization (SDR) method to solve the p-center problem.…”

Section: Time Optimization Base-station Selectionmentioning

confidence: 99%

Optimal Multi-Base-Station Selection for Belt-Area Wireless Sensor Networks

et al. 2019

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Since underground environments, such as urban subways, tunnels, underground pipe corridors, and mine roadways, etc., are complex and changeable, the software functions of the sensor nodes in underground spaces must be updated regularly to satisfy the requirements of the monitoring center. Software updates cannot be traditionally conducted due to the large number of nodes, the wide distribution range and the poor underground environment; they can only be made using wireless reprogramming. Current reprogramming protocols are all designed for topologically unconstrained networks and are inefficient in routing constrained underground belt-area wireless sensor networks (BAWSNs). We propose a new reprogramming mechanism for reducing the energy and time consumption of the data downstream transmission process in BAWSNs. For network energy optimization, we identified the highest energy efficiency transmission radius. For network base-station location time optimization, an approximate (1 + ε) algorithm that is based on gradient cyclic descent is proposed, which is of complexity O kn 2. The simulation results demonstrate that, compared with classical algorithms, the BAWSNs approximation algorithm can locate the optimal base station accurately with low time consumption. INDEX TERMS Belt-area wireless sensor networks, base station selection, p-center problem, optimization algorithm.

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When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem

Cited by 12 publications

References 16 publications

Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem

Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem

Approximation Algorithms for the Vertex K-Center Problem: Survey and Experimental Evaluation

Optimal Multi-Base-Station Selection for Belt-Area Wireless Sensor Networks

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