Geodesic
Some Solvability Classes for the Problem of Integer Balancing of a Three-Dimensional Matrix with Constraints of Second Type
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 54-69.

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The problem of integer balancing of a three-dimensional matrix with constraints of second type is studied. The elements of the inner part (all three indices are greater than zero) of the three-dimensional matrix are summed in each direction and each section of the matrix; the total sum is also found. These sums are placed into the elements where one or more indices are equal to zero (according to the summing directions). The problem is to find an integer matrix of the same structure, which can be produced from the initial one by replacing the elements of the inner part with the largest previous or the smallest following integer. At the same time variations of the sums of elements from that in the initial matrix should be less than 2 and the element with three zero indices should be produced with standard rules of rounding-off. Some solvability classes for this problem are defined. Also, a model of reducing this problem to a problem of finding the maximum flow in a multiple network and an algorithm for the corresponding flow problem are suggested. A polynomial algorithm for the particular case of $n=2$ is described.
Keywords: integer balancing, three-dimensional matrices, constraints of second type, solvability classes, multiple networks, multiple flows, generalized labeling algorithm. 
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A. V. Smirnov. Some Solvability Classes for the Problem of Integer Balancing of a Three-Dimensional Matrix with Constraints of Second Type. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 2, pp. 54-69. http://geodesic.mathdoc.fr/item/MAIS_2013_20_2_a3/

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