Given a set of connection requests (calls) in a communication network, the call control problem is to accept a subset of the requests and route them along paths in the network such that no edge capacity is violated, with the goal of rejecting as few requests as possible. We investigate the complexity of parameterized versions of this problem, where the number of rejected requests is taken as the parameter. For the variant with pre-determined paths, the problem is fixed-parameter tractable in arbitrary graphs if the link capacities are bounded by a constant, but W-hard if the link capacities are arbitrary. If the paths are not pre-determined, no FPT algorithm can exist even in series–parallel graphs unless P =NP. Our main results are new FPT algorithms for call control in tree networks with arbitrary edge capacities and in trees of rings with unit edge capacities in the case that the paths are not pre-determined
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