# Network Model for the Problem of Integer Balancing of a Four-Dimensional Matrix

## Abstract

The problem of integer balancing of a four-dimensional matrix is studied. The elements of the inner part (all four indices are greater than zero) of the given real matrix are summed in each direction and each two- and three-dimensional 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 with the largest previous or the smallest following integer. At the same time, the element with four zero indices should be produced with standard rules of rounding-off. In the article the problem of finding the maximum multiple flow in the network of any natural multiplicity *k* is also studied. There are arcs of three types: ordinary arcs, multiple arcs and multi-arcs. Each multiple and multi-arc is a union of *k* linked arcs, which are adjusted with each other. The network constructing rules are described. The definitions of a divisible network and some associated subjects are stated. There are defined the basic principles for reducing the integer balancing problem of an *l*-dimensional matrix (*l* ≥ 3) to the problem of finding the maximum flow in a divisible multiple network of multiplicity *k*. There are stated the rules for reducing the four-dimensional balancing problem to the maximum flow problem in the network of multiplicity 5. The algorithm of finding the maximum flow, which meets the solvability conditions for the integer balancing problem, is formulated for such a network.

### Keywords

integer balancing multiple networks multiple flows divisible networks*NP*-completeness generalized labeling algorithm four-dimensional matrices

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