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Embedding signed graphs in the line

Author

Listed:
  • Eduardo G. Pardo

    (Universidad Rey Juan Carlos)

  • Mauricio Soto

    (Université d’Orléans)

  • Christopher Thraves

    (Universidad Rey Juan Carlos)

Abstract

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388–399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

Suggested Citation

  • Eduardo G. Pardo & Mauricio Soto & Christopher Thraves, 2015. "Embedding signed graphs in the line," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 451-471, February.
    • Handle: RePEc:spr:jcomop:v:29:y:2015:i:2:d:10.1007_s10878-013-9604-1
    DOI: 10.1007/s10878-013-9604-1
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    References listed on IDEAS

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    1. Loiola, Eliane Maria & de Abreu, Nair Maria Maia & Boaventura-Netto, Paulo Oswaldo & Hahn, Peter & Querido, Tania, 2007. "A survey for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 176(2), pages 657-690, January.
    2. Harary, Frank & Kabell, Jerald A., 1980. "A simple algorithm to detect balance in signed graphs," Mathematical Social Sciences, Elsevier, vol. 1(1), pages 131-136, September.
    3. Zvi Drezner & Peter Hahn & Éeric Taillard, 2005. "Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for Meta-Heuristic Methods," Annals of Operations Research, Springer, vol. 139(1), pages 65-94, October.
    4. Jadranka Skorin-Kapov, 1990. "Tabu Search Applied to the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 2(1), pages 33-45, February.
    5. Thomas A. Feo & Mauricio G. C. Resende & Stuart H. Smith, 1994. "A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set," Operations Research, INFORMS, vol. 42(5), pages 860-878, October.
    6. Burkard, R. E. & Rendl, F., 1984. "A thermodynamically motivated simulation procedure for combinatorial optimization problems," European Journal of Operational Research, Elsevier, vol. 17(2), pages 169-174, August.
    7. Connolly, David T., 1990. "An improved annealing scheme for the QAP," European Journal of Operational Research, Elsevier, vol. 46(1), pages 93-100, May.
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    Cited by:

    1. Eduardo G. Pardo & Antonio García-Sánchez & Marc Sevaux & Abraham Duarte, 2020. "Basic variable neighborhood search for the minimum sitting arrangement problem," Journal of Heuristics, Springer, vol. 26(2), pages 249-268, April.
    2. Sergio Cavero & Eduardo G. Pardo & Abraham Duarte, 2022. "A general variable neighborhood search for the cyclic antibandwidth problem," Computational Optimization and Applications, Springer, vol. 81(2), pages 657-687, March.
    3. Micheal Arockiaraj & Jessie Abraham & Arul Jeya Shalini, 2019. "Node set optimization problem for complete Josephus cubes," Journal of Combinatorial Optimization, Springer, vol. 38(4), pages 1180-1195, November.
    4. Julio Aracena & Christopher Thraves Caro, 2023. "The weighted sitting closer to friends than enemies problem in the line," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-21, January.

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