Local Search Algorithms for the Vertex K-Center Problem

Díaz, Jesús García; Mendez, Ricardo Menchaca; Hernandez, Jairo Sanchez; Méndez, Rolando Menchaca

doi:10.1109/tla.2018.8444397

Cited by 14 publications

(6 citation statements)

References 11 publications

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“…Original Graph (G) 1 st Iteration: Graph is connected and hence it has a single component. Backbone path bbpath = [16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1] and best central node is 10 for radius 3. Here, by radius we mean the number of remaining steps in the burning sequence.…”

Section: Resultsmentioning

confidence: 99%

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Faster heuristics for graph burning

2021

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Graph burning is a process of information spreading through the network by an agent in discrete steps. The problem is to find an optimal sequence of nodes that have to be given information so that the network is covered in least number of steps. Graph burning problem is NP-Hard for which two approximation algorithms and a few heuristics have been proposed in the literature. In this work, we propose three heuristics, namely, Backbone Based Greedy Heuristic (BBGH), Improved Cutting Corners Heuristic (ICCH), and Component Based Recursive Heuristic (CBRH). These are mainly based on Eigenvector centrality measure. BBGH finds a backbone of the network and picks vertex to be burned greedily from the vertices of the backbone. ICCH is a shortest path based heuristic and picks vertex to burn greedily from best central nodes. The burning number problem on disconnected graphs is harder than on the connected graphs. For example, burning number problem is easy on a path where as it is NP-Hard on disjoint paths. In practice, large networks are generally disconnected and moreover even if the input graph is connected, during the burning process the graph among the unburned vertices may be disconnected. For disconnected graphs, ordering the components is crucial. Our CBRH works well on disconnected graphs as it prioritizes the components. All the heuristics have been implemented and tested on several bench-mark networks including large networks of size more than 50K nodes. The experimentation also includes comparison to the approximation algorithms. The advantages of our algorithms are that they are much simpler to implement and also several orders faster than the heuristics proposed in the literature.

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Section: Resultsmentioning

confidence: 99%

“…There are a few problems related to graph burning proposed in the literature. K -centre problem [ 11 ], is one in which the K centres are chosen simultaneously as sources of fire. As a result the nodes burn in parallel and hence very quickly.…”

Section: Introductionmentioning

confidence: 99%

Faster heuristics for graph burning

2021

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“…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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“…The length of the optimal burning sequence is called the burning number of the graph, denoted by b(G). The Graph Burning problem is similar to the k-centre problem [6], in which all the sources of fire are selected at the first step itself.…”

Section: Introductionmentioning

confidence: 99%

Improved Approximation Algorithm for Graph Burning on Trees

Gautam¹,

Kare²,

S³

2022

Preprint

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Given a graph G = (V, E), the problem of Graph Burning is to find a sequence of nodes from V , called burning sequence, in order to burn the whole graph. This is a discrete-step process, in each step an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A node that is burnt spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The Graph Burning problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a 3-approximation algorithm for general graphs and a 2-approximation algorithm for trees. In this paper, we propose an approximation algorithm for trees that produces a burning sequence of length at most 1.75b(T ) + 1, where b(T ) is length of the optimal burning sequence, also called the burning number of the tree T . In other words, we achieve an approximation factor of ( 1.75b(T ) + 1)/b(T ).

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Local Search Algorithms for the Vertex K-Center Problem

Cited by 14 publications

References 11 publications

Faster heuristics for graph burning

Faster heuristics for graph burning

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

Improved Approximation Algorithm for Graph Burning on Trees

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