Abstract
This paper is devoted to a classical NP-hard problem, known as the three-index axial assignment problem. Within the corresponding framework, the problem of combining feasible solutions is posed as an assignment problem on the set of solutions containing only the components of selected feasible solutions. The issues of combining solutions for the multicriteria problem with different criteria convolutions are studied. In the general case, the combination problem turns out to be NP-hard. Polynomial solvability conditions are obtained for the combination problem.
REFERENCES
Spieksma, F.C.R., Multi Index Assignment Problems. Complexity, Approximation, Applications, in Nonlinear Assignment Problems: Algorithms and Applications, Pardalos, P.M. and Pitsoulis, L.S., Eds., Dordrecht: Kluwer Acad. Publishers, 2000, pp. 1–11.
Burkard, R., Dell’Amico, M., and Martello, S., Assignment Problems, Philadelphia: SIAM, 2012.
Kuroki, Y. and Matsui, T., An Approximation Algorithm for Multidimensional Assignment Problems Minimizing the Sum of Squared Errors, Discret. Appl. Math., 2009, vol. 157, no. 9, pp. 2124–2135.
Poore, A.B., Multidimensional Assignment Problems Arising in Multitarget and Multisensor Tracking, in Nonlinear Assignment Problems: Algorithms and Applications, Pardalos, P.M. and Pitsoulis, L.S., Eds., Dordrecht: Kluwer Acad. Publishers, 2000, pp. 13–38.
Garey, M.R. and Johnson, D.S., Computers and Intractability. A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979.
Crama, Y. and Spieksma, F.C.R., Approximation Algorithms for Three-Dimensional Assignment Problems with Triangle Inequalities, Eur. J. Oper. Res., 1992, vol. 60, pp. 273–279.
Bandelt, H.J., Crama, Y., and Spieksma, F.C.R., Approximation Algorithms for Multidimensional Assignment Problems with Decomposable Costs, Discret. Appl. Math., 1994, vol. 49, pp. 25–50.
Burkard, R.E., Rudolf, R., and Woeginger, G.J., Three-Dimensional Axial Assignment Problems with Decomposable Cost Coefficients, Discret Appl. Math., 1996, vol. 65, pp. 123–139.
Spieksma, F. and Woeginger, G., Geometric Three-Dimensional Assignment Problems, Eur. J. Oper. Res., 1996, vol. 91, pp. 611–618.
Custic, A., Klinz, B., and Woeginger, G.J., Geometric Versions of the Three-Dimensional Assignment Problem under General Norms, Discret. Optim., 2015, vol. 18, pp. 38–55.
Balas, E. and Saltzman, M.J., An Algorithm for the Three-Index Assignment Problem, Oper. Res., 1991, vol. 39, no. 1, pp. 150–161.
Natu, S., Date, K., and Nagi, R., GPU-Accelerated Lagrangian Heuristic for Multidimensional Assignment Problems with Decomposable Costs, Parallel Comput., 2020, vol. 97, art. no. 102666.
Huang, G. and Lim, A., A Hybrid Genetic Algorithm for the Three-Index Assignment Problem, Eur. J. Oper. Res., 2006, vol. 172, pp. 249–257.
Kim, B.J., Hightower, W.L., Hahn, P.M., Zhu, Y.R., and Sun, L., Lower Bounds for the Axial Three-Index Assignment Problem, Eur. J. Oper., 2010, vol. 202, pp. 654–668.
Dichkovskaya, S.A. and Kravtsov, M.K., Investigation of Polynomial Algorithms for Solving the Three-Index Planar Assignment Problem, Comput. Math. and Math. Phys., 2006, vol. 46, no. 2, pp. 212–217.
Dichkovskaya, S.A. and Kravtsov, M.K., Investigation of Polynomial Algorithms for Solving the Multi-criteria Three-Index Planar Assignment Problem, Comput. Math. and Math. Phys., 2007, vol. 47, no. 6, pp. 1029–1038.
Emelichev, V.A. and Perepelitsa, V.A., Complexity of Discrete Multicriteria Problems, Discrete Math. Appl., 1994, vol. 4, no. 2, pp. 89–118.
Prilutskii, M.Kh., Multicriterial Multi-index Resource Scheduling Problems, J. Comput. Syst. Sci. Int., 2007, vol. 46, no. 1, pp. 78–82.
Prilutskij, M.Kh., Multicriteria Distribution of a Homogeneous Resource in Hierarchical Systems, Autom. Remote Control, 1996, vol. 57, no. 2, part 2, pp. 266–271.
Afraimovich, L.G. and Emelin, M.D., Combining Solutions of the Axial Assignment Problem, Autom. Remote Control, 2021, vol. 82, no. 8, pp. 1418–1425.
Afraimovich, L.G. and Emelin, M.D., Heuristic Strategies for Combining Solutions of the Three-Index Axial Assignment Problem, Autom. Remote Control, 2021, vol. 82, no. 10, pp. 1635–1640.
Afraimovich, L.G. and Emelin, M.D., Complexity of Solutions Combination for the Three-Index Axial Assignment Problem, Mathematics, 2022, vol. 10, no. 7, p. 1062.
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Afraimovich, L.G., Emelin, M.D. Criteria Convolutions When Combining the Solutions of the Multicriteria Axial Assignment Problem. Autom Remote Control 85, 718–726 (2024). https://doi.org/10.1134/S0005117924700152
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DOI: https://doi.org/10.1134/S0005117924700152