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Criteria Convolutions When Combining the Solutions of the Multicriteria Axial Assignment Problem

  • OPTIMIZATION, SYSTEM ANALYSIS, AND OPERATIONS RESEARCH
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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.

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REFERENCES

  1. 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.

    Google Scholar 

  2. Burkard, R., Dell’Amico, M., and Martello, S., Assignment Problems, Philadelphia: SIAM, 2012.

  3. 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.

    Article  MathSciNet  Google Scholar 

  4. 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.

    Google Scholar 

  5. Garey, M.R. and Johnson, D.S., Computers and Intractability. A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979.

    Google Scholar 

  6. 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.

    Article  Google Scholar 

  7. 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.

    Article  Google Scholar 

  8. 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.

    Article  MathSciNet  Google Scholar 

  9. Spieksma, F. and Woeginger, G., Geometric Three-Dimensional Assignment Problems, Eur. J. Oper. Res., 1996, vol. 91, pp. 611–618.

    Article  Google Scholar 

  10. 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.

    Article  MathSciNet  Google Scholar 

  11. Balas, E. and Saltzman, M.J., An Algorithm for the Three-Index Assignment Problem, Oper. Res., 1991, vol. 39, no. 1, pp. 150–161.

    Article  MathSciNet  Google Scholar 

  12. 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.

  13. 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.

    Article  MathSciNet  Google Scholar 

  14. 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.

    Article  MathSciNet  Google Scholar 

  15. 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.

    Article  MathSciNet  Google Scholar 

  16. 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.

    Article  MathSciNet  Google Scholar 

  17. Emelichev, V.A. and Perepelitsa, V.A., Complexity of Discrete Multicriteria Problems, Discrete Math. Appl., 1994, vol. 4, no. 2, pp. 89–118.

    Article  MathSciNet  Google Scholar 

  18. Prilutskii, M.Kh., Multicriterial Multi-index Resource Scheduling Problems, J. Comput. Syst. Sci. Int., 2007, vol. 46, no. 1, pp. 78–82.

    Article  MathSciNet  Google Scholar 

  19. 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.

  20. 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.

    Article  MathSciNet  Google Scholar 

  21. 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.

    Article  MathSciNet  Google Scholar 

  22. 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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Correspondence to L. G. Afraimovich or M. D. Emelin.

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This paper was recommended for publication by A.A. Lazarev, a member of the Editorial Board

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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

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