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General variable neighborhood search for the minimum stretch spanning tree problem

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Abstract

Given an undirected graph, the minimum stretch spanning tree problem (MSSTP) deals with finding a spanning tree such that the maximum distance in the tree for adjacent nodes in the original graph, called the stretch, is minimum. This is an NP-hard problem with many applications in transportation and communication networks. We propose a general variable neighborhood search (GVNS) algorithm based on a balance between solution generation and improvement. To achieve this balance, we consider different construction heuristics and neighborhood strategies to efficiently explore the search space. To assess the merit of our proposal, we perform extensive experimentation on various classes of graphs consisting of 214 instances. A comparison in terms of solution quality and execution time with the best previous method, namely an artificial bee colony (ABC) algorithm, shows the superiority of GVNS. Results are compared using statistical tests to draw significant conclusions.

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Acknowledgements

The first author would like to thank University Grants Commission (UGC), India for providing RGNF fellowship [RGNF-2014-15-SC-UTT-74975] during the research. The research by Rafael Martí is funded by “Ministerio de Ciencia, Innovación y Universidades” of Spain under grant refs. PGC2018-0953322-B-C21 and PID2021-125709OB-C21 MCIU/AEI/FEDER-UE.

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Authors and Affiliations

  1. Department of Mathematics, Dayalbagh Educational Institute, Agra, India

    Yogita Singh Kardam & Kamal Srivastava

  2. Department of Statistics and Operations Research, University of Valencia, Valencia, Spain

    Rafael Martí

Authors
  1. Yogita Singh Kardam
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  2. Kamal Srivastava
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  3. Rafael Martí
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Correspondence to Kamal Srivastava.

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Appendix A

Appendix A

Note- In all the tables the results of instances for which the performance of GVNS is same as that of ABC or ABC_Nbr are shown in bold whereas the values in bold with asterisk show the improvement of our algorithm over ABC and ABC_Nbr.

See Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

Table 6 Comparison of results obtained by ABC, ABC_Nbr and GVNS for Wheel Graphs
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Table 7 Comparison of results obtained by ABC, ABC_Nbr and GVNS for complete Graphs
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Table 8 Comparison of results obtained by ABC, ABC_Nbr and GVNS for Split Graphs
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Table 9 Comparison of results obtained by ABC, ABC_Nbr and GVNS for complete k-Partite Graphs
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Table 10 Comparison of results obtained by ABC, ABC_Nbr and GVNS for Triangular Grids
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Table 11 Comparison of results obtained by ABC, ABC_Nbr and GVNS for Rectangular Grids
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Table 12 Comparison of results obtained by ABC, ABC_Nbr and GVNS for Triangulated Rectangular Grids
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Table 13 Comparison of results obtained by ABC, ABC_Nbr and GVNS for small HB graphs
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Table 14 Comparison of results obtained by ABC, ABC_Nbr and GVNS for medium HB graphs
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Table 15 Comparison of results obtained by ABC, ABC_Nbr and GVNS for large HB graphs
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Kardam, Y.S., Srivastava, K. & Martí, R. General variable neighborhood search for the minimum stretch spanning tree problem. Optim Lett 17, 2005–2031 (2023). https://doi.org/10.1007/s11590-022-01918-1

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