Geodesic
The problem of integer-valued balancing of a three-dimensional matrix and algorithms of its solution
Modelirovanie i analiz informacionnyh sistem, Tome 17 (2010) no. 2, pp. 72-98

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The article is devoted to the problem of integer-valued balancing of a three-dimensional matrix. The reduction of this problem to the problem of finding a maximum flow in the multiple network of integer-valued balancing and the algorithm for this problem are suggested. Also, the comparative characteristic of two algorithms of integer-valued balancing is made according to the results of the computing experiments. NP-completeness of the problem of integer-valued balancing of a three-dimensional matrix is proved in the article. The problem of minimization of the errors of rounding off in the problem of integer-valued balancing is explored.
Keywords: integer-valued balancing, three-dimensional matrices, multiple networks, multiple flows, generalized labeling algorithm, first Gomory algorithm, $NP$-completeness, minimization of the errors of rounding off. 
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     title = {The problem of integer-valued balancing of a three-dimensional matrix and algorithms of its solution},
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V. S. Rublev; A. V. Smirnov. The problem of integer-valued balancing of a three-dimensional matrix and algorithms of its solution. Modelirovanie i analiz informacionnyh sistem, Tome 17 (2010) no. 2, pp. 72-98. http://geodesic.mathdoc.fr/item/MAIS_2010_17_2_a4/