Shortest Path and Maximum Flow Problems Under Service Function Chaining Constraints

Abstract

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

Robust network function virtualization

“…SFC route generation and NF forwarding graphs embedding are the two approaches to manage SFC requests either individually or jointly. Hence, even with the simplest case, the minimal weighted SFC path problem is different from the shortest path problem, which requires visiting VNF instantiated physical nodes in the desired sequence.…”

“…Even et al provide a randomized approximation algorithm leveraging multicommodity flows for path computation and function placement. Sallam et al focus on the SFC counterpart of standard graph theory problems, such as the minimal weighted SFC path and the SFC maximal flow. The proposed pseudo‐polynomial algorithms solve the minimal weighted SFC paths on a transferred network graph and a special case of SFC maximal flow problem.…”

“…We first propose a new evaluation metric on NF‐service reliability under the worst‐case scenario, followed by studying robust NF provisioning under the failure uncertainty of NF‐enabled node. As mentioned earlier, two subproblems are involved in the NF provisioning problem: (1) the minimal‐weighted SFC path problem: different from , we first propose a Dijkstra‐like algorithm with an extended level of information; and (2) robust NF provisioning: we present an approximation algorithm for SFC‐Fork and a mixed‐integer programming formulation as its exact solution approach for the general cases.…”

Placement and Allocation of Virtual Network Functions: Multi-dimensional Case

“…In NFV, several studies (e.g., [4], [13], [14]) consider the placement of a minimum number of VNF instances to cover all flows. A single type of network functions is considered in [4], [13], [15], [16], and the case of multiple network functions is considered in [14], [17], [18], [19], [20]. However, these work neglects either the budget constraint or the multi-dimensional resource allocation.…”

Online Joint Placement and Allocation of Virtual Network Functions With Heterogeneous Servers

“…Vast works on NFV are done under the SFC constraints. For example, several such interesting problems are considered by Sallam et al [15]: 1) How to find an SFCconstrained shortest path between any pair of nodes? 2) What is the achievable SFC-constrained maximum flow?…”