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BVNS for the Minimum Sitting Arrangement Problem in a Cycle

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Variable Neighborhood Search (ICVNS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13863))

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Abstract

Signed graphs are a particular type of graph, in which the vertices are connected through edges labeled with a positive or negative weight. Given a signed graph, the Minimum Sitting Arrangement (MinSA) problem aims to minimize the total number of errors produced when the graph (named the input graph) is embedded into another graph (named the host graph). An error in this context appears every time that a vertex with two adjacent (one positive and one negative) is embedded in such a way that the adjacent vertex connected with a negative edge is closer than the adjacent vertex connected with a positive edge. The MinSA can be used to model a variety of real-world problems, such as links in online social networks, the location of facilities, or the relationships between a group of people. Previous studies of this problem have been mainly focused on host graphs, whose structure is a path. However, in this research, we compare two variants of the MinSA that differ in the graph used as the host graph (i.e., a path or a cycle). Particularly, we adapted a previous state-of-the-art algorithm for the problem based on Basic Variable Neighborhood Search. The solutions obtained are compared with the solutions provided by a novel Branch & Bound algorithm for small instances. We also analyze the differences found by using non-parametric statistical tests.

This research has been partially supported by grants: PID2021-125709OA-C22, PID2021-126605NB-I00 and FPU19/0409, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”; grant P2018/TCS-4566, funded by Comunidad de Madrid and European Regional Development Fund; grant CIAICO/2021/224 funded by Generalitat Valenciana; and grant M2988 funded by Proyectos Impulso de la Universidad Rey Juan Carlos 2022. We would also like to thank authors of [21] for sharing their code with us.

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Authors and Affiliations

  1. Universidad Rey Juan Carlos, Madrid, Spain

    Marcos Robles, Sergio Cavero & Eduardo G. Pardo

Authors
  1. Marcos Robles
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  2. Sergio Cavero
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  3. Eduardo G. Pardo
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Corresponding author

Correspondence to Sergio Cavero .

Editor information

Editors and Affiliations

  1. Khalifa University, Abu Dhabi, United Arab Emirates

    Andrei Sleptchenko

  2. University of Macedonia, Thessaloniki, Greece

    Angelo Sifaleras

  3. HEC Montréal, Montréal, QC, Canada

    Pierre Hansen

Appendix: Individual results per instance

Appendix: Individual results per instance

Next, we present the individual results per instance obtained by each algorithm. In the case of the B &B procedure, results obtained before 3600 s (when the method is truncated) correspond to optimal solutions (Tables 3, 4 and 5).

Table 3. Complete instances.
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Table 4. Interval instances.
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Table 5. Random instances.
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Robles, M., Cavero, S., Pardo, E.G. (2023). BVNS for the Minimum Sitting Arrangement Problem in a Cycle. In: Sleptchenko, A., Sifaleras, A., Hansen, P. (eds) Variable Neighborhood Search. ICVNS 2022. Lecture Notes in Computer Science, vol 13863. Springer, Cham. https://doi.org/10.1007/978-3-031-34500-5_7

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  • DOI: https://doi.org/10.1007/978-3-031-34500-5_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34499-2

  • Online ISBN: 978-3-031-34500-5

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