Optimization of an SMD placement machine and flows in parametric networks

In the minimization of the number of subtours made by an SMD placement machine the following variant of the minimum cost flow problem arised: in a network with n vertices and m arcs a set F of arcs (parametrized arcs) is given and the cost of the flow is equal to the maximum of flow values along the arcs from F . This problem can be solved by a sequence of maximum flow computations in modified networks where the capacities of the parametrized arcs are successively set to an increasing sequence of target parameter values. We show that it suffices to consider at most O(m) different target values and so this approach leads to a strongly polynomial algorithm consisting of at most O(m) maximum flow computations.