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Multiindex transportation problems with 2-embedded structure

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Abstract

We consider multiindex transportation problems of linear and integer linear programming. As a method of solving them, we propose an approach based on reductions of multiindex transportation problems to min-flow problems. We show that under the reduction scheme we consider, the 2-embeddability condition for multiindex problems is a necessary and sufficient condition for the problem to be reducible to a min-cost flow problem.

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References

  1. Afraimovich, L.G. and Prilutskii, M.Kh., Multiindex Resource Distributions for Hierarchical Systems, Autom. Remote Control, 2006, vol. 67, no. 6, pp. 1007–1016.

    Article  MATH  Google Scholar 

  2. Afraimovich, L.G. and Prilutskii, M.Kh., Multiproduct Flows in Tree-Like Networks, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2008, no. 2, pp. 57–63.

  3. Prilutskii, M.Kh., Multicriterial Multiindex Problems of Volume Calendar Planning, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2007, no. 1, pp. 78–82.

  4. Prilutskii, M.Kh., Multicriterial Distribution of a Homogeneous Resource in Hierarchical Systems, Autom. Remote Control, 1996, vol. 57, no. 2, part 2, pp. 266–271.

    Google Scholar 

  5. Kostyukov, V.E. and Prilutskii, M.Kh., Resource Distribution in Hierarchical Systems. Optimization Problems for Mining and Transportation of Gas and Refining Gas Condensate, Nizhni Novgorod: Nizhegorod. Gos. Univ., 2010.

    Google Scholar 

  6. Lim, A., Rodrigues, B., and Zhang, X., Scheduling Sports Competitions at Multiple Venues-Revisited, Eur. J. Oper. Res., 2006, vol. 175, pp. 171–186.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gunawan, A., Ng, K.M., and Poh, K.L., Solving the Teacher Assignment-Course Scheduling Problem by a Hybrid Algorithm, Int. J. Comput. Inform. Eng., 2007, vol. 1, no. 2, pp. 137–142.

    Google Scholar 

  8. Storms, P.P.A. and Spieksma, F.C.R., An LP-Based Algorithm for the Data Association Problem in Multitarget Tracking, Comput. Oper. Res., 2003, vol. 30, no. 7, pp. 1067–1085.

    Article  MathSciNet  MATH  Google Scholar 

  9. Poore, A.B., Multidimensional Assignment Formulation of Data Association Problems Arising from Multitarget and Multisensor Tracking, Comput. Optim. Appl., 1994, vol. 3, no. 1, pp. 27–57.

    Article  MathSciNet  MATH  Google Scholar 

  10. Schrijver, A., Theory of Linear and Integer Programming, New York: Wiley, 1986. Translated under the title Teoriya lineinogo i tselochislennogo programmirovaniya, Moscow: Mir, 1991.

    MATH  Google Scholar 

  11. Papadimitrou, Ch. and Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs: Prentice Hall, 1982. Translated under the title Kombinatornaya optimizatsiya. Algoritmy i slozhnost’, Moscow: Mir, 1985.

    Google Scholar 

  12. Gale, D., The Theory of Linear Economic Models, New York: McGraw-Hill, 1960. Translated under the title Teoriya lineinykh ekonomicheskikh modelei, Moscow: Mir, 1969.

    Google Scholar 

  13. Raskin, L.G. and Kirichenko, I.O., Mnogoindeksnye zadachi lineinogo programmirovaniya (Multiindex Linear Programming Problems), Moscow: Radio i Svyaz’, 1982.

    Google Scholar 

  14. Emelichev, V.A., Kovalev, M.M., and Kravtsov, M.K., Mnogogranniki, grafy, optimizatsiya (Polyhedra, Graphs, and Optimization), Moscow: Nauka, 1981.

    MATH  Google Scholar 

  15. Bandopadhyaya, L. and Puri, M.C., Impaired Flow Multi-Index Transportation Problem with Axial Constraints, J. Austral. Math. Soc., Ser. B, 1988, vol. 29(3), pp. 296–309.

    Article  MathSciNet  Google Scholar 

  16. Junginger, W., On Representatives of Multi-Index Transportation Problems, Eur. J. Oper. Res., 1993, vol. 66(3), pp. 353–371.

    Article  MATH  Google Scholar 

  17. De Loera, J.A., Kim, E.D., Onn, S., and Santos, F., Graphs of Transportation Polytopes, J. Combinat. Theory, Ser. A, 2009, vol. 116, no. 8, pp. 1306–1325.

    Article  MATH  Google Scholar 

  18. Kravtsov, M.A. and Krachkovskii, A.P., On Certain Properties of Three-Index Transportation Polytopes, Diskret. Mat., 1999, vol. 11, no. 3, pp. 109–125.

    MathSciNet  Google Scholar 

  19. Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NPCompleteness, San Francisco: Freeman, 1979. Translation under the title Vychislitel’nye mashiny i trudnoreshaemye zadachi, Moscow: Mir, 1982.

    Google Scholar 

  20. 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  MATH  Google Scholar 

  21. Finkel’shtein, Yu.Yu., Priblizhennye metody i prikladnye zadachi diskretnogo programmirovaniya (Approximate Methods and Applied Problems of Discrete Programming), Moscow: Nauka, 1976.

  22. Spieksma, F.C.R., Multi Index Assignment Problems: Complexity, Approximation, Applications, in Nonlinear Assignment Problems: Algorithms and Applications, P.M. Pardalos, L.S. Pitsoulis, Eds., Dordrecht: Kluwer, 2000, pp. 1–11.

    Google Scholar 

  23. Gimadi, E.Kh. and Korkishko, N.M., On One Algorithm of Solving a Three-Index Axial Assignment Problem on Single-Cycle Permutations, Diskret. Anal. Issled. Oper., Ser. 1, 2003, vol. 10, no. 2, pp. 56–65.

    MathSciNet  MATH  Google Scholar 

  24. Gimadi, E.Kh. and Glazkov, Yu.V., On an Asymptotically Exact Algorithm for Solving One Modification of the Three-Index Planar Assignment Problem, Diskret. Anal. Issled. Oper., Ser. 2, 2006, vol. 13, no. 1, pp. 10–26.

    MathSciNet  MATH  Google Scholar 

  25. Sergeev, S.I., New Lower Bounds for the Triplanar Assignment Problem. Use of the Classical Model, Autom. Remote Control, 2008, vol. 69, no. 12, pp. 2039–2060.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ahuja, R.K., Magnati, T.L., and Orlin, J.B., Network Flows: Theory, Algorithms, and Applications, New Jersey: Prentice Hall, 1993.

    MATH  Google Scholar 

  27. Orlin, J.B., A Faster Strongly Polynomial Minimum Cost Flow Algorithm, Oper. Res., 1993, vol. 41, no. 2, pp. 338–350.

    Article  MathSciNet  MATH  Google Scholar 

  28. Goldberg, A.V. and Rao, S., Beyond the Flow Decomposition Barrier, J. ACM, 1998, vol. 45, no. 5, pp. 783–797.

    Article  MathSciNet  MATH  Google Scholar 

  29. Litvak, B.G. and Rappoport, A.M., Linear Programming Problems That Allow a Network Formulation, Ekonom. Mat. Metody, 1970, vol. 6, no. 4, pp. 594–604.

    MathSciNet  Google Scholar 

  30. Lin, Y., A Recognition Problem in Converting Linear Programming to Network Flow Models, Appl. Math. J. Chinese Univ., 1993, vol. 8, no. 1, pp. 76–85.

    Article  MATH  Google Scholar 

  31. Kovalev, M.M., Matroidy v diskretnoi optimizatsii (Matroid in Discrete Optimization), Moscow: Editorial URSS, 2003.

    Google Scholar 

  32. Gülpinar, N., Gutin, G., Mitra, G., and Zverovitch, A., Extracting Pure Network Submatrices in Linear Programs using Signed Graphs, Discret. Appl. Math., 2004, vol. 137, no. 3, pp. 359–372.

    Article  MATH  Google Scholar 

  33. Afraimovich, L.G., Three-Index Linear Programs with Nested Structure, Autom. Remote Control, 2011, vol. 72, no. 8, pp. 1679–1689.

    Article  MathSciNet  MATH  Google Scholar 

  34. Afraimovich, L.G., Cyclic Reducibility of Multiindex Systems of Linear Inequalities of Transportation Type, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2010, no. 4, pp. 83–90.

  35. Chen, B., Potts, C.N., and Woeginger, G.J., A Review of Machine Scheduling. Complexity, Algorithms and Approximability, in Handbook of Combinatorial Optimization, New York: Kluwer, 1998, vol. 3, pp. 21–169.

    Google Scholar 

  36. Gofman, A.D. and Kraskal, D.B., Integer-Valued Boundary Points of Convex Polyhedra, in Lineinye neravenstva i smezhnye voprosy (Linear Inequalities and Adjacent Problems), Moscow: Inostrannaya Literatura, 1959, pp. 325–347.

    Google Scholar 

  37. Katerov, A.S., A Study of the Reducibility of Three-Index Transportation Problems to Finding a Flow in a Network with Parallel Computations, in Prikladnaya informatika i matematicheskoe modelirovanie (Aplied Informatics and Mathematical Modeling), Moscow: MGUP, 2011, pp. 47–57.

    Google Scholar 

  38. Prilutskii, M.Kh., Decomposition of a Homogeneous Resource in Hierarchical Systems with Tree-Like Structure, in Proc. Int. Conf. “Systems Identification and Control problems” (SICPRO’2000), Moscow, September 26–28, 2000, Moscow: Inst. Probl. Upravlen., 2000, pp. 2038–2049.

    Google Scholar 

  39. Borradaile, G., Klein, P.N., Mozes, S., Nussbaum, Y., and Wulff-Nilsen, S., Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time, CoRR, abs/1105.2228, 2011.

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Authors and Affiliations

  1. Nizhni Novgorod State University, Nizhni Novgorod, Russia

    L. G. Afraimovich

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Original Russian Text © L.G. Afraimovich, 2013, published in Avtomatika i Telemekhanika, 2013, No. 1, pp. 116–134.

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Afraimovich, L.G. Multiindex transportation problems with 2-embedded structure. Autom Remote Control 74, 90–104 (2013). https://doi.org/10.1134/S0005117913010086

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