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

Abstract

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.

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

“…In addition to the classical formulations F1U and F1C, many alternative integer programming or mixed integer programming formulations have been proposed for the capacitated and uncapacitated vertex k-center problem [5,6,[26][27][28][29][30][31][32]. Perhaps, the simplest of these formulations are based on the relationship between the uncapacitated vertex k-center problem and the NP-Hard set covering and minimum dominating set problems [26,35,36]. In fact, the uncapacitated vertex k-center problem is equivalent to the minimum dominating set problem when the size r(C * ) of the optimal solution C * is known ahead of time (Lemma 1 and Theorem 1) [35,36].…”

“…Perhaps, the simplest of these formulations are based on the relationship between the uncapacitated vertex k-center problem and the NP-Hard set covering and minimum dominating set problems [26,35,36]. In fact, the uncapacitated vertex k-center problem is equivalent to the minimum dominating set problem when the size r(C * ) of the optimal solution C * is known ahead of time (Lemma 1 and Theorem 1) [35,36]. To better understand this relationship, Definition 1 describes what a dominating set is and Definition 2 describes what a minimum dominating set is.…”

“…To better understand this relationship, Definition 1 describes what a dominating set is and Definition 2 describes what a minimum dominating set is. Then, Expressions (16) to (19) show the integer programming formulation for the uncapacitated vertex k-center as a minimum dominating set problem [11,[36][37][38].…”

“…This way, by selecting random capacities from the range[ 0.75 • n−k k , 1.25 • n−k k], the capacity of each vertex will be relatively close to n−k k . This is important because, as the capacity of all vertices approaches n, the problem becomes closer to the uncapacitated version, which is easier to solve according to empirical experiments[32,36]. Thus, the generated instances remain relatively difficult to solve.…”

Approximation Algorithm-Based Prosumer Scheduling for Microgrids

“…Algorithm 3 is motivated by the idea of transforming the prosumer scheduling problem into the k-center problem, then using the algorithms in [32,33] to obtain the minimum operational cost, where k is the number of suppliers in the optimal solution. The best possible polynomial-time algorithm for the k-center problem currently is to provide 2-approximated solutions; finding a solution in the well-known NP-Hard problems, such as the minimum dominating-set problem.…”