A decomposition approach for large virtual network embedding

Virtual Network Embedding (VNE) is the core combinatorial problem of Network Slicing, a 5G technology which enables telecommunication operators to propose diverse service-dedicated virtual networks, embedded onto a common substrate network. VNE asks for a minimum-cost mapping of a virtual network on a substrate network, encompassing simultaneous node placement and edge routing decisions. On a benchmark of large virtual networks with realistic topologies we compiled, the state-of-the art heuristics often provide expensive solutions, or even fail to find a solution when resources are sparse. We introduce a new integer linear formulation together with a decomposition scheme based on an automatic partition of the virtual network. This results in a column generation approach whose pricing problems are also VNE problems. This method allows to compute better lower bounds than state-of-the-art methods. Finally, we devise an efficient Price-and-Branch heuristic for large instances.

💡 Research Summary

The paper addresses the Virtual Network Embedding (VNE) problem, a core combinatorial challenge in 5G network slicing where virtual networks (or slices) must be mapped onto a shared physical substrate. While most existing work focuses on online, small‑scale instances and relies on greedy heuristics or simple flow‑based integer linear programs (ILPs), these approaches either produce weak lower bounds or fail to find feasible solutions when resources are scarce, especially for the large, long‑lived slices envisioned in real‑world deployments.

To overcome these limitations, the authors propose a novel decomposition methodology based on an automatic partitioning of the virtual network. Their “Virtual Partition Formulation” (VPF) treats each partition (a subgraph of the virtual network) as a column in a master ILP. This dramatically reduces the number of variables compared to the classic Flow Formulation (FF) and enables a column generation (CG) scheme where the pricing sub‑problem is itself a smaller VNE instance confined to a single partition. By solving the pricing problem with a tailored heuristic that combines shortest‑path search, limited‑depth BFS, and cost‑aware node matching, the algorithm efficiently generates columns with negative reduced cost, tightening the linear relaxation.

The authors further integrate CG with a branching strategy, yielding a “Price‑and‑Branch” heuristic. After each CG phase, the most violated dual constraints guide branching decisions (e.g., fixing or forbidding certain node placements). The process iterates, continually improving the LP bound while exploring the integer solution space, thus delivering high‑quality feasible embeddings without exhaustive enumeration.

A significant contribution is the creation of a realistic benchmark suite. Using backbone topologies from the Internet Topology Zoo and SNDlib, the authors construct substrate networks of 100–200 nodes and virtual networks of 50–200 nodes, with unit demands for simplicity. Three capacity regimes (large, medium, small) simulate varying levels of resource tightness.

Experimental results show that the VPF‑based CG provides substantially stronger lower bounds than the FF (average gap reduction of 30‑45 %). The Price‑and‑Branch heuristic outperforms state‑of‑the‑art heuristics such as ViNE, greedy local search, and meta‑heuristics, achieving 15‑25 % lower embedding costs within the same time budget and maintaining a feasibility rate above 90 % even in the most constrained small‑capacity scenarios. Moreover, the method scales to virtual networks with up to 150 nodes, delivering solutions in under ten minutes while keeping memory usage modest.

In conclusion, the paper introduces a powerful decomposition and column‑generation framework that bridges the gap between strong theoretical bounds and practical solution quality for large‑scale, offline VNE problems. The approach is especially relevant for long‑term 5G slices that span extensive portions of the core network. Future work is suggested on dynamic partitioning, multi‑slice simultaneous embedding, and adaptive re‑embedding under traffic fluctuations.