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

Acosta, Jose; García-Díaz, Jesús; Menchaca-Mendez, Rolando; Menchaca-Mendez, Rolando

doi:10.3390/math8091551

Cited by 5 publications

(5 citation statements)

References 39 publications

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“…Although the VKCP and the GBP are different, their approximation algorithms are conceptually similar [7,8,19]. This similarity comes from the fact that the VKCP has a polynomial-time reduction to the minimum dominating set problem, which can be viewed as the problem of burning all vertices in parallel in one single step [19][20][21][22]. Regarding the FP, it aims at protecting vertices from burning given an initial set of fire sources [23][24][25].…”

Section: Definitionmentioning

confidence: 99%

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Graph Burning: Mathematical Formulations and Optimal Solutions

et al. 2022

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The graph burning problem is an NP-hard combinatorial optimization problem that helps quantify how vulnerable a graph is to contagion. This paper introduces three mathematical formulations of the problem: an integer linear program (ILP) and two constraint satisfaction problems (CSP1 and CSP2). Thanks to off-the-shelf optimization software, these formulations can be solved optimally over arbitrary graphs; this is relevant because the only algorithms designed to date for this problem are approximation algorithms and heuristics, which do not guarantee to find optimal solutions. We empirically compared the proposed formulations using random graphs and off-the-shelf optimization software. The results show that CSP1 and CSP2 tend to reach optimal solutions in less time than the ILP. Therefore, we executed them over some benchmark graphs of order at most 5908. The previously best-known solutions for some of these graphs were improved. We draw some empirical observations from the experimental results. For instance, we find the tendency: the larger the graph’s optimal solution, the more difficult it is to find it. Finally, the resulting set of optimal solutions might be helpful as a benchmark dataset for the performance evaluation of non-exact algorithms.

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Section: Definitionmentioning

confidence: 99%

“…• Finally, Equations ( 19) and (20) indicate that all vertices are unburned at the beginning and that the burning sequence begins as an empty sequence.…”

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confidence: 99%

Graph Burning: Mathematical Formulations and Optimal Solutions

et al. 2022

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“…Next, it is worth mentioning the so-called minimum total dominating set problem, which was solved by means of a hybrid evolutionary algorithm in [19]. Finally, it is also worth mentioning that high-quality algorithms for the CapMDS problem can also be useful for solving problems, such as the capacitated vertex k-center problem; see [20].…”

Section: Literature Review On Related Problemsmentioning

confidence: 99%

Barrakuda: A Hybrid Evolutionary Algorithm for Minimum Capacitated Dominating Set Problem

Pinacho-Davidson

Blum

2020

Mathematics

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The minimum capacitated dominating set problem is an NP-hard variant of the well-known minimum dominating set problem in undirected graphs. This problem finds applications in the context of clustering and routing in wireless networks. Two algorithms are presented in this work. The first one is an extended version of construct, merge, solve and adapt, while the main contribution is a hybrid between a biased random key genetic algorithm and an exact approach which we labeled BARRAKUDAmathsizesmall. Both algorithms are evaluated on a large set of benchmark instances from the literature. In addition, they are tested on a new, more challenging benchmark set of larger problem instances. In the context of the problem instances from the literature, the performance of our algorithms is very similar. Moreover, both algorithms clearly outperform the best approach from the literature. In contrast, BARRAKUDAmathsizesmall is clearly the best-performing algorithm for the new, more challenging problem instances.

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“…This problem is being actively studied, as are various algorithms for solving it [9]- [11]. Different variants and versions of the problem are described in the scientific literature [12]- [15]. Some of these variants are based on non-smooth optimization techniques [13], others are related to restricted covering problem [16] and fixed-parameter approximations in transportation networks [17], [18], as well as those applied in trees [19].…”

Section: Introductionmentioning

confidence: 99%

“…In this case, the distances obtained after generating all k-element subsets C of V are checked. There are different algorithms for solving the problem, both exact [15] and approximate [17], [30].…”

Section: Introductionmentioning

confidence: 99%

An analysis between exact and approximate algorithms for the k-center problem in graphs

Kralev

Kraleva

Ankov

et al. 2022

IJECE

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<span lang="EN-US">This research focuses on the k-center problem and its applications. Different methods for solving this problem are analyzed. The implementations of an exact algorithm and of an approximate algorithm are presented. The source code and the computation complexity of these algorithms are presented and analyzed. The multitasking mode of the operating system is taken into account considering the execution time of the algorithms. The results show that the approximate algorithm finds solutions that are not worse than two times optimal. In some case these solutions are very close to the optimal solutions, but this is true only for graphs with a smaller number of nodes. As the number of nodes in the graph increases (respectively the number of edges increases), the approximate solutions deviate from the optimal ones, but remain acceptable. These results give reason to conclude that for graphs with a small number of nodes the approximate algorithm finds comparable solutions with those founds by the exact algorithm.</span>

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

Cited by 5 publications

References 39 publications

Graph Burning: Mathematical Formulations and Optimal Solutions

Graph Burning: Mathematical Formulations and Optimal Solutions

Barrakuda: A Hybrid Evolutionary Algorithm for Minimum Capacitated Dominating Set Problem

An analysis between exact and approximate algorithms for the k-center problem in graphs

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