A rapid learning automata-based approach for generalized minimum spanning tree problem
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A rapid learning automata-based approach for generalized minimum spanning tree problem Masoumeh Zojaji1 · Mohammad Reza Mollakhalili Meybodi1 · Kamal Mirzaie1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Generalized minimum spanning tree problem, which has several real-world applications like telecommunication network designing, is related to combinatorial optimization problems. This problem belongs to the NP-hard class and is a minimum tree on a clustered graph spanning one node from each cluster. Although exact and metaheuristic algorithms have been applied to solve the problems successfully, obtaining an optimal solution using these approaches and other optimization tools has been a challenge. In this paper, an attempt is made to achieve a sub-optimal solution using a network of learning automata (LA). This algorithm assigns an LA to every cluster so that the number of actions is the same as that of nodes in the corresponding cluster. At each iteration, LAs select one node from their clusters. Then, the weight of the constructed generalized spanning tree is considered as a criterion for rewarding or penalizing the selected actions. The experimental results on a set of 20 benchmarks of TSPLIB demonstrate that the proposed approach is significantly faster than the other mentioned algorithms. The results indicate that the new algorithm is competitive in terms of solution quality. Keywords Generalized minimum spanning tree · Induced subgraph · Learning automata · Combinatorial optimization problems · Random search
1 Introduction Generalized minimum spanning tree (GMST) was employed to support various problems in the field of graph optimization and has received much attention over the past 30 years (Dror and Haouari 2000). In a clustered graph, GMST aims to find a minimum spanning tree in which every node belongs to only one cluster. Although GMST and the minimum spanning tree (MST) have similar behavior, when the problem is solved
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Masoumeh Zojaji [email protected] Department of Computer Engineering, Maybod Branch, Islamic Azad University, Maybod, Iran
123
Journal of Combinatorial Optimization
Fig. 1 A clustered graph with 20 nodes Fig. 2 An array of selected node numbers
Fig. 3 A generalized spanning tree
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by well-known polynomial algorithms of Prim and Kruskal (Golden et al. 2005), the inclusion of the concept of clustering and the selection of a node from each cluster, turns GMST into an NP-hard problem (Myung and Lee 1995). A weighted, undirected graph G in principle can be represented as G(V , E) where V is a set of nodes and E is a set of edges connecting node pairs in V . In a ’weighted graph’, each edge includes additional information called weights and the Euclidean distance between two nodes can be considered as the weight of the corresponding edge. Let R be an equivalence relation partitioning the node-set V into
123
Journal of Combinatorial Optimization
M equivalent classes called clusters. Figure 1 shows a clustered graph with 2
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