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SDP-based bounds for graph partition via extended ADMM

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  • Angelika Wiegele

    (Alpen-Adria-Universität Klagenfurt)

  • Shudian Zhao

    (Alpen-Adria-Universität Klagenfurt)

Abstract

We study two NP-complete graph partition problems, k-equipartition problems and graph partition problems with knapsack constraints (GPKC). We introduce tight SDP relaxations with nonnegativity constraints to get lower bounds, the SDP relaxations are solved by an extended alternating direction method of multipliers (ADMM). In this way, we obtain high quality lower bounds for k-equipartition on large instances up to $$n =1000$$ n = 1000 vertices within as few as 5 min and for GPKC problems up to $$n=500$$ n = 500 vertices within as little as 1 h. On the other hand, interior point methods fail to solve instances from $$n=300$$ n = 300 due to memory requirements. We also design heuristics to generate upper bounds from the SDP solutions, giving us tighter upper bounds than other methods proposed in the literature with low computational expense.

Suggested Citation

  • Angelika Wiegele & Shudian Zhao, 2022. "SDP-based bounds for graph partition via extended ADMM," Computational Optimization and Applications, Springer, vol. 82(1), pages 251-291, May.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:1:d:10.1007_s10589-022-00355-1
    DOI: 10.1007/s10589-022-00355-1
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    References listed on IDEAS

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    1. Neng Fan & Panos Pardalos, 2010. "Linear and quadratic programming approaches for the general graph partitioning problem," Journal of Global Optimization, Springer, vol. 48(1), pages 57-71, September.
    2. Dirk A. Lorenz & Quoc Tran-Dinh, 2019. "Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence," Computational Optimization and Applications, Springer, vol. 74(1), pages 67-92, September.
    3. David S. Johnson & Cecilia R. Aragon & Lyle A. McGeoch & Catherine Schevon, 1989. "Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning," Operations Research, INFORMS, vol. 37(6), pages 865-892, December.
    4. Bissan Ghaddar & Miguel Anjos & Frauke Liers, 2011. "A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem," Annals of Operations Research, Springer, vol. 188(1), pages 155-174, August.
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