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With the advent of Network Function Virtualization (NFV), Physical NetworkFunctions (PNFs) are gradually being replaced by Virtual Network Functions(VNFs) that are hosted on general purpose servers. Depending on the call flowsfor specific services, the packets need to pass through an ordered set ofnetwork functions (physical or virtual) called Service Function Chains (SFC)before reaching the destination. Conceivably for the next few years during thistransition, these networks would have a mix of PNFs and VNFs, which brings aninteresting mix of network problems that are studied in this paper: (1) How tofind an SFC-constrained shortest path between any pair of nodes? (2) What isthe achievable SFC-constrained maximum flow? (3) How to place the VNFs suchthat the cost (the number of nodes to be virtualized) is minimized, while themaximum flow of the original network can still be achieved even under the SFCconstraint? In this work, we will try to address such emerging questions.First, for the SFC-constrained shortest path problem, we propose atransformation of the network graph to minimize the computational complexity ofsubsequent applications of any shortest path algorithm. Second, we formulatethe SFC-constrained maximum flow problem as a fractional multicommodity flowproblem, and develop a combinatorial algorithm for a special case of practicalinterest. Third, we prove that the VNFs placement problem is NP-hard andpresent an alternative Integer Linear Programming (ILP) formulation. Finally,we conduct simulations to elucidate our theoretical results.

Shortest Path and Maximum Flow Problems Under Service Function Chaining Constraints