Time-slot assignment for TDMA-systems

In satellite communication as in other technical systems using the TDMA-technique (time division multiple access), the problem arises to decompose a given (n x n)-matrix in a weighted sum of permutation matrices such that the sum of the weights becomes minimal. It is shown at first that an optimal solution of this problem can be obtained in O (n exp 4)-time using at most O (n-squared) different permutation matrices. Thereafter a short discussion in given of the decomposition in O(n) different matrices which turns out to be NP-hard. Moreover, it is shown that an optimal decomposition using a class of 2 n permutation matrices which are fixed in advance can be obtained by solving a classical assignment problem. This latter problem can be generalized by taking arbitrary Boolean matrices. The corresponding decomposition problem can be transformed to a special max flow min cost network flow problem, and is thus soluble by a genuinely polynomial algorithm.