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A dual approach to multi-dimensional assignment problems

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Abstract

In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency.

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References

  1. Abdallah, F., Nassreddine, G., Denœux, T.: A multiple-hypothesis map-matching method suitable for weighted and box-shaped state estimation for localization. IEEE Trans. Intell. Transp. Syst. 12, 1495–1510 (2011)

    Article  Google Scholar 

  2. Balinski, M.L.: Signature methods for the assignment problem. Oper. Res. 33, 527–536 (1985)

    Article  MathSciNet  Google Scholar 

  3. Bandelt, H.J., Crama, Y., Spieksma, F.C.: Approximation algorithms for multi-dimensional assignment problems with decomposable costs. Discret. Appl. Math. 49, 25–50 (1994)

    Article  MathSciNet  Google Scholar 

  4. Bar-Shalom, Y., Willett, P.K., Tian, X.: Tracking and Data Fusion: A Handbook of Algorithms. YBS Publishing, Storrs (2011)

    Google Scholar 

  5. Bertsekas, D.P., Castanon, D.A.: A forward/reverse auction algorithm for asymmetric assignment problems. Comput. Optim. Appl. 1, 277–279 (1992)

    Article  MathSciNet  Google Scholar 

  6. Blackman, S.S.: Multiple hypothesis tracking for multiple target tracking. IEEE Aerosp. Electron. Syst. Mag. 19, 5–18 (2004)

    Article  Google Scholar 

  7. Burkard, R.E.: Selected topics on assignment problems. Discret. Appl. Math. 123, 257–302 (2002)

    Article  MathSciNet  Google Scholar 

  8. Deb, S., Yeddanapudi, M., Pattipati, K.R., Bar-Shalom, Y.: Generalized S-D assignment algorithm for multisensor-multitarget state estimation. IEEE Trans. Aerosp. Electron. Syst. 33, 523–538 (1997)

    Article  Google Scholar 

  9. Dell’Amico, M., Toth, P.: Algorithms and codes for dense assignment problems: the state of the art. Discret. Appl. Math. 100, 17–48 (2000)

    Article  MathSciNet  Google Scholar 

  10. Emami, P., Pardalos, P.M., Elefteriadou, L., Ranka, S.: Machine learning methods for solving assignment problems in multi-target tracking. arXiv preprint arXiv:1802.06897 (2018)

  11. Frieze, A.M., Yadegar, J.: An algorithm for solving 3-dimensional assignment problems with application to scheduling a teaching practice. J. Oper. Res. Soc. 32, 989–995 (1981)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  13. Garg, V., Wickramarathne, T.: MHT approach to ubiquitous monitoring of spatio-temporal phenomena. In: 21st International Conference on Information Fusion, Cambridge, UK, pp. 1816–1821 (2018)

  14. Grundel, D.A., Pardalos, P.M.: Test problem generator for the multidimensional assignment problem. Comput. Optim. Appl. 30, 133–146 (2005)

    Article  MathSciNet  Google Scholar 

  15. Jonker, R., Volgenant, A.: A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38, 325–340 (1987)

    Article  MathSciNet  Google Scholar 

  16. Karapetyan, D., Gutin, G.: Local search heuristics for the multidimensional assignment problem. J. Heuristics 17, 201–249 (2011)

    Article  Google Scholar 

  17. Kirubarajan, T., Bar-Shalom, Y., Pattipati, K.R., Kadar, I.: Ground target tracking with variable structure IMM estimator. IEEE Trans. Aerosp. Electron. Syst. 36, 26–46 (2000)

    Article  Google Scholar 

  18. Kirubarajan, T., Bar-Shalom, Y., Pattipati, K.R.: Multiassignment for tracking a large number of overlapping objects. IEEE Trans. Aerosp. Electron. Syst. 37, 2–21 (2001)

    Article  Google Scholar 

  19. Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Log. Quart. 2, 83–97 (1955)

    Article  MathSciNet  Google Scholar 

  20. Lawler, E.L., Wood, D.E.: Branch-and-bound methods: a survey. Oper. Res. 14, 699–719 (1966)

    Article  MathSciNet  Google Scholar 

  21. Lau, R.A., Williams, J.L.: Multidimensional assignment by dual decomposition. In: Seventh International Conference on Intelligent Sensors, Sensor Networks and Information Processing, Adelaide, Australia, pp. 437–442 (2011)

  22. Li, J., Tharmarasa, R., Brown, D., et al.: A novel convex dual approach to three-dimensional assignment problem: theoretical analysis. Comput. Optim. Appl. 74, 481–516 (2019). https://doi.org/10.1007/s10589-019-00113-w

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J.: Efficient data association algorithms for multi-target tracking. PhD thesis, McMaster University. Sep., 2019. On line access: http://hdl.handle.net/11375/25001 (2019)

  24. Pattipati, K.R., Deb, S., Bar-Shalom, Y., Washburn, R.B.: A new relaxation algorithm and passive sensor data association. IEEE Trans. Autom. Control 37, 198–212 (1992)

    Article  Google Scholar 

  25. Pentico, D.W.: Assignment problems: a golden anniversary survey. Eur. J. Oper. Res. 176, 774–793 (2007)

    Article  MathSciNet  Google Scholar 

  26. Plitsos, S., Magos, D., Mourtos, I.: An integrated solver for multi-index assignment. In: International Symposium on Artificial Intelligence and Mathematics, Fort Lauderdale, FL (2016)

  27. Poore, A.B., Rijavec, N.: A Lagrangian relaxation algorithm for multidimensional assignment problems arsing from multitarget tracking. SIAM J. Optim. 3, 544–563 (1993)

    Article  MathSciNet  Google Scholar 

  28. Poore, A.B., Robertson III, A.J.: A new Lagrangian relaxation based algorithm for a class of multidimensional assignment problems. Comput. Optim. Appl. 8, 129–150 (1997)

    Article  MathSciNet  Google Scholar 

  29. Poore, A.B., Gadaleta, S.: Some assignment problems arising from multiple target tracking. Math. Comput. Modell. 43, 1074–1091 (2006)

    Article  MathSciNet  Google Scholar 

  30. Pattipati, K., Popp, R., Kirubarajan, T.: Survey of assignment techniques for multitarget tracking. In: Bar-Shalom, Y., Blair, W.D. (eds.) Multitarget-Multisensor Tracking: Applications and Advances, vol. III, p. 2000. Artech House, Dedham (2000)

    Google Scholar 

  31. Robertson, A.J.: A set of greedy randomized adaptive local search procedure (GRASP) implementations for the multidimensional assignment problem. Comput. Optim. Appl. 19, 145–164 (2001)

    Article  MathSciNet  Google Scholar 

  32. Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer, Berlin (1985)

    Book  Google Scholar 

  33. Vogiatzis, C., Pasiliao, E.L., Pardalos, P.M.: Graph partitions for the multidimensional assignment problem. Comput. Optim. Appl. 58, 205–224 (2014)

    Article  MathSciNet  Google Scholar 

  34. Walteros, J.L., Vogiatzis, C., Pasiliao, E.L., Pardalos, P.M.: Integer programming models for the multidimensional assignment problem with star costs. Eur. J. Oper. Res. 235, 553–568 (2014)

    Article  MathSciNet  Google Scholar 

  35. Wang, H., Kirubarajan, T., Bar-Shalom, Y.: Large scale air traffic surveillance using IMM estimators with assignment. IEEE Trans. Aerosp. Electron. Syst. 35, 255–266 (1999)

    Article  Google Scholar 

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Acknowledgements

(1) We would like to thank the two anonymous reviewers for their comments and suggestions to improve the paper. (2) The paper was originally taken from the PhD thesis work of the first author, and later was revised after he lost his job in the pandemic and supported by the Canada Emergency Response Benefit (CERB). It would be impossible to complete this work without the CERB support to which the first author is deeply grateful.

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Jingqun Li: Currently unemployed: looking for a machine learning position for conducting research on combinatorial optimization, computer vision and natural language processing.

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Li, J., Kirubarajan, T., Tharmarasa, R. et al. A dual approach to multi-dimensional assignment problems. J Glob Optim 81, 691–716 (2021). https://doi.org/10.1007/s10898-020-00988-8

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