We study consistent migration of flows, with special focus on software defined networks. Given a current and a desired network flow configuration, we give the first polynomial-time algorithm to decide if a congestion-free migration is possible. However, if all flows must be integer or are unsplittable, this is NP-hard to decide. A similar problem is providing increased bandwidth to an application, while keeping all other flows in the network, but possibly migrating them consistently to other paths. We show that the maximum increase can be approximated arbitrarily well in polynomial time. Current methods as RSVP-TE consider unsplittable flows and remove flows of lesser importance in order to increase bandwidth for an application: We prove that deciding what flows need to be removed is an NP-hard optimization problem with no PTAS possible unless P = NP.

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