[Paper Review] Shortest Path and Maximum Flow Problems Under Service Function Chaining Constraints

Gamal Sallam, Gagan Gupta+2 more|arXiv (Cornell University)|Jan 17, 2018

Software-Defined Networks and 5G10 references3 citations

TL;DR

This paper addresses shortest path and maximum flow problems under Service Function Chaining (SFC) constraints in hybrid networks with both physical and virtual network functions (PNFs/VNFs). It proposes a graph transformation technique to efficiently compute SFC-constrained shortest paths, formulates the SFC-constrained maximum flow as a fractional multicommodity flow problem, and presents an ILP formulation for VNF placement that minimizes the number of nodes while preserving original maximum flow, proving the problem NP-hard and demonstrating low-cost placement in simulations.

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.

Motivation & Objective

• To address the challenge of finding shortest paths that satisfy ordered service function chains (SFCs) in hybrid networks with both PNFs and VNFs.
• To determine the maximum feasible flow under SFC constraints, especially when new VNFs are introduced during NFV deployment.
• To minimize the number of network nodes where VNFs are placed while preserving the original maximum flow capacity.
• To develop computationally efficient and scalable solutions for SFC-constrained network optimization problems in real-world hybrid network topologies.

Proposed method

• Transform the original network graph G into a layered graph G̅ to encode SFC constraints, enabling standard shortest path algorithms to compute SFC-constrained paths efficiently.
• Introduce a pruning algorithm to reduce the size of the transformed graph, improving computational efficiency for repeated queries.
• Formulate the SFC-constrained maximum flow problem as a fractional multicommodity flow problem solvable via linear programming.
• Propose a combinatorial algorithm based on Ford-Fulkerson for a special case of SFC-MF where all flows must pass through a single newly deployed VNF.
• Develop an Integer Linear Programming (ILP) formulation for the VNF placement problem to minimize the number of nodes hosting VNFs while maintaining original maximum flow.
• Integrate the SFC-constrained shortest path algorithm into a throughput-optimal routing framework to achieve stability and high utilization.

Experimental results

Research questions

• RQ1 How can we efficiently compute the shortest path between two nodes when the path must traverse a specific sequence of network functions (SFC)?
• RQ2 What is the maximum flow that can be achieved between a source and destination when all packets must pass through a given ordered set of network functions?
• RQ3 What is the minimum number of network nodes required to place VNFs such that the original maximum flow capacity is preserved under SFC constraints?
• RQ4 Can a combinatorial algorithm be designed for the general SFC-constrained maximum flow problem, or is LP-based solution necessary?
• RQ5 How does the placement of VNFs affect network throughput and queue stability in the presence of SFC constraints?

Key findings

• The proposed graph transformation and pruning technique significantly reduces the size of the transformed graph, especially when function availability is high, leading to faster shortest path computation.
• The SFC-constrained maximum flow value may be strictly less than the original maximum flow, depending on the placement of network functions in the network.
• The VNF placement problem is proven to be NP-hard, confirming the computational difficulty of minimizing placement cost under SFC constraints.
• The ILP-based VNF placement solution achieves the original maximum flow with a small number of nodes—e.g., only one node (v6) was sufficient to maintain a maximum flow of 8 in a 10-node network.
• Simulations show that integrating the SFC-constrained shortest path into a throughput-optimal routing framework maintains finite average queue lengths when arrival rates are within the capacity region.
• For random graphs of size 10 to 100 nodes, the average number of VNF placement nodes required to preserve maximum flow remains low, indicating cost-effective deployment potential.