NP-Completeness of the Integer Balancing Problem for a Three-Dimensional Matrix

NP-completeness of the integer balancing problem for a three-dimensional matrix is studied. The integer balancing problem for a three-dimensional matrix is formulated and a three-dimensional matrix with rational elements that satisfies the balance conditions is considered. A matrix schedule of transporting empty cars, which are grouped according to certain characteristics, is also considered. The original matrix is used to construct a network with certain vertices assigned to the matrix elements and with the capacities of the network arcs assigned to these matrix elements. It is shown that the classical NP-complete problem of 3-dimensional matching can be reduced to IB3 in a polynomial time. The individual 3MM problem has a solution if and only if the IB3M problem obtained by solving the system of linear inequalities for the given set in the 3MM problem has a solution.