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On the Embed and Project Algorithm for the Graph Bandwidth Problem

Author

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  • Janez Povh

    (Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva ulica 6, SI-1000 Ljubljana, Slovenia
    Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia)

Abstract

The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000, which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n ≤ 1000 .

Suggested Citation

  • Janez Povh, 2021. "On the Embed and Project Algorithm for the Graph Bandwidth Problem," Mathematics, MDPI, vol. 9(17), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2030-:d:620783
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    References listed on IDEAS

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    1. E. R. van Dam & R. Sotirov, 2015. "On Bounding the Bandwidth of Graphs with Symmetry," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 75-88, February.
    2. Marti, Rafael & Campos, Vicente & Pinana, Estefania, 2008. "A branch and bound algorithm for the matrix bandwidth minimization," European Journal of Operational Research, Elsevier, vol. 186(2), pages 513-528, April.
    3. Alberto Caprara & Juan-José Salazar-González, 2005. "Laying Out Sparse Graphs with Provably Minimum Bandwidth," INFORMS Journal on Computing, INFORMS, vol. 17(3), pages 356-373, August.
    4. Fischer, I. & Gruber, G. & Rendl, F. & Sotirov, R., 2006. "Computational experience with a bundle approach for semidenfinite cutting plane relaxations of max-cut and equipartition," Other publications TiSEM 03dfd8c3-9216-4c75-8921-3, Tilburg University, School of Economics and Management.
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