Multicommodity network flow with jump constraints

Multicommodity network flow is an important problem that occurs in many areas of operations research. Given a set of commodities which are to be carried between certain nodes of a capacitate network, the general problem is to find an optimal distribution of the traffic such that all the needs are satisfied without violating any capacity constraint. This problem can be formulated as a large linear program whose structure can be used to speed up the simplex method [ 1, 4, 5, 71: price directive and resource directive decomposition, partitioning, specific technique for the GUB structure, etc. The most promising adaptation of the simplex method designed to solve multicommodity flow problem seems to be the primal partitioning technique for the arc-chain formulation introduced by Farvolden et al. [3]. The aim of the present paper is to show how the same approach can be used to solve a little more complicated problem, arising from telecommunication network optimization: the multicommodity flow problem with ,jump constraints. This problem differs from the previous one by the fact that commodities can only flow through paths with no more than a fixed number of arcs. Furthermore, a new result and shortest proofs dealing with properties of solutions produced by primal partitioning are presented. We also note that it is possible to extend the same partitioning idea to the node-arc formulation of the problem but without taking into account the jump constraints.