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A product form representation of the inverse of a multicommodity cycle matrix

When solving multicommodity network flow problems with either a primal or a dual partitioning technique one must carry and update a working basis inverse whose size need never exceed the number of saturated arcs (i.e. arcs for which there is no excess capacity). Efficient procedures have appeared in the literature for updating this inverse if it is carried in explicit form. However, for real world multicommodity network flow problems, this inverse may become quite large. In addition, this matrix may be quite dense even though the working basis may be sparse. In an attempt to obtain a sparse representation of this inverse matrix and simplify the computational burden, we present a new data structure along with the updating formulae for storing and updating this matrix. This data structure is a generalization of the well-known product form procedure frequently used in mathematical programming codes.