Virtual Network Embedding Polytope

• VNE Polytope is the convex hull of incidence vectors from feasible virtual network embeddings, capturing the combinatorial structure of network design.
• New valid inequalities—flow-departure, flow-continuity, and leaf-equalities—are introduced to tighten mixed-integer programming relaxations.
• A tailored flow-decomposition algorithm on path substrates guarantees cycle-free mappings and demonstrates the polytope's integrality and computational benefits.

The Virtual Network Embedding (VNE) polytope is the convex hull of incidence vectors corresponding to feasible embeddings of a virtual network into a substrate network, with the embedding constraints modeled through an undirected "flow" formulation. This object encapsulates the combinatorial structure of feasible solutions to the VNE problem, which is fundamental in modern telecommunication network design and optimization. The polyhedral study of the VNE polytope aims to establish tight descriptions using valid inequalities and to clarify when the associated mixed-integer programming relaxations are integral.

1. Flow Formulation and VNE Polytope Definition

The undirected flow formulation rigorously specifies the mapping of a virtual graph $G_r=(V_r, E_r)$—where nodes $\bar u\in V_r$ and edges $\bar e\in E_r$ have demands $d_{\bar u}\in\mathbb Z_+$, $d_{\bar e}\in\mathbb Z_+$—onto a substrate graph $G_s=(V_s, E_s)$ with capacities $c_u\in\mathbb Z_+$ for nodes, $c_e\in\mathbb Z_+$ for edges, and costs $w_u, w_e\ge0$. The substrate is bidirected via $G'_s = (V_s, E'_s)$, in which each undirected $(u,v)\in E_s$ is replaced by arcs $(u,v),(v,u)\in E'_s$.

Assignment variables are $x_{\bar u,u} \in \{0,1\}$, indicating placement of virtual node $\bar u$ at substrate $u$, and $y_{\bar e,(u,v)} \in \{0,1\}$, indicating utilization of arc $(u,v)$ for virtual edge $\bar e=(\bar u, \bar v)$. The flow formulation (FF) comprises node-placement constraints, flow-conservation, capacity, and integrality constraints: $$\begin{aligned} & \sum_{u\in V_s} x_{\bar u, u} = 1 && \forall \bar u \in V_r \ & x_{\bar u, u} - x_{\bar v, u} = \sum_{(u,v)\in\delta^+(u)} y_{\bar e, (u,v)} - \sum_{(v,u)\in\delta^-(u)} y_{\bar e, (v,u)} && \forall \bar e=(\bar u,\bar v)\in E_r,\; \forall u\in V_s \ & \sum_{\bar u\in V_r} x_{\bar u, u} \leq 1 && \forall u\in V_s \ & x_{\bar u, u} \in \{0,1\};\quad y_{\bar e, (u,v)} \in \{0,1\} && \forall \bar u, u, \bar e, (u,v) \ & \sum_{\bar e\in E_r} d_{\bar e}(y_{\bar e,(u,v)} + y_{\bar e,(v,u)}) \le c_{\{u,v\}} && \forall\{u,v\}\in E_s \end{aligned}$$

The VNE polytope is

$$\mathcal F = \operatorname{conv}\left\{ \chi(m) \mid m \text{ is a feasible mapping} \right\} \subseteq \mathbb{R}^{x, y}$$

where $\chi(m)$ denotes the incidence vector of an embedding.

2. Families of Valid Inequalities

The polyhedral description is substantially strengthened by introducing new valid inequalities:

• Flow-Departure Inequalities: For all $\bar e=(\bar u,\bar v)\in E_r$ and $u\in V_s$,

$$x_{\bar u,u} \le \sum_{(u,v)\in\delta^+(u)} y_{\bar e,(u,v)}$$

This requires that if $\bar u$ is placed on $u$, the embedding of $\bar e$ must depart $u$.

• Flow-Continuity Inequalities: For all $\bar e=(\bar u,\bar v)\in E_r$ and $(u,v)\in E'_s$,

$$y_{\bar e,(u,v)} \le \sum_{\substack{(v,w)\in\delta^+(v)\ w\neq u}} y_{\bar e,(v,w)} + x_{\bar v,v}$$

If $\bar e$ enters $v$ via $(u,v)$, the flow must either terminate at $\bar v$ placed at $v$ or continue along another outgoing arc.

• Leaf-Equalities: For each leaf $\ell \in L_s$ (leaves of substrate tree) and its sole neighbor $v_\ell$,

$y_{\bar e,(\ell,v_\ell)} = x_{\bar u,\ell}$

Ensuring that virtual nodes mapped to leaves induce flow on the unique incident arc.

All three families of inequalities are valid for $\mathcal F$; they tighten the relaxation and are critical for full characterization in certain substrate topologies.

3. Flow-Decomposition Algorithm for Path Substrates

For the special case where $E_r=\{\bar e\}$ is a singleton and $G_s$ is a simple path $u_1-\cdots-u_{n_s}$, feasible fractional solutions to (FF)+(A)+(B) can be decomposed as convex combinations of cycle-free $\bar u$–$\bar v$ paths. Unlike classical flow decomposition, which may generate cycles, a bespoke algorithm—ComputeFlowDecomposition—ensures that only cycle-free paths corresponding to feasible VNE mappings are produced. The algorithm iteratively extracts forward and backward paths, maintaining feasibility of the residuals for (A)+(B) and terminating when all $x, y$ variables vanish. Each extraction corresponds to a valid embedding, and the weights sum to unity.

The algorithmic key properties are:

• Residuals retain feasibility for all required constraints after every extraction.
• The process does not generate cycles; all paths are source-target embeddings.
• Upon completion, the solution is a convex combination of integer-valued mappings.

Thus, on path substrates with a single virtual edge, every fractional extreme point of the feasible region corresponds to an integral solution.

4. Polyhedral Characterization and Integrality Results

A formal facet characterization is established for the VNE polytope under the single-edge, path-substrate case. The integral property holds: if every feasible fractional solution admits a cycle-free path decomposition, the associated polytope is integral. The principal result is:

$$\mathcal F(\{\bar e\},\,\text{path }u_1\!-\!\cdots-\!u_{n_s}) = \Bigl\{ (x, y) \in [0,1] \,\Big|\, \text{(FF) eqs. (1)–(5)}, \text{ (A) flow-departure}, \text{ (B) flow-continuity} \Bigr\}$$

In this regime, all extreme points of the relaxation are integral. The cut families (A), (B), and (when appropriate) (C) define facets of the relaxed VNE polytope and, jointly with (FF), precisely describe its structure for the setting of a single virtual edge embedded on a substrate path.

5. Computational Experiments

Computational experiments implemented the basic (FF) formulation with and without the new inequality families in CPLEX 22.1, using random Erdős–Rényi virtual graphs ($n_r=14$, $m_r=22$) and substrate topologies from Topology Zoo (70–75 nodes). Node capacities varied: 25%, 50%, or 100% of nodes assigned $c_u=1$, all edges unit capacity, uniform random costs. Over 100 instances, the average computational time, branch-and-bound nodes, and LP relaxation value $v(\text{LP})$ were measured.

Node-cap. %	Constraints	Time (s)	#B&B nodes	$v(\text{LP})$
25%	FF	46.5	11565	29.6
	FF + FD	22.4	7441	72.1
	FF + FD + FC	24.6	4741	86.4
50%	FF	107.0	25639	20.4
	FF + FD	38.4	12684	51.8
	FF + FD + FC	65.1	9124	52.7
100%	FF	211.1	38125	14.7
	FF + FD	106.6	16377	40.9
	FF + FD + FC	224.4	12340	40.9

Key findings:

• Inclusion of flow-departure (FD) inequalities nearly halves average CPU time and branch-and-bound tree size, while significantly improving LP bounds.
• Adding flow-continuity (FC) inequalities further improves the bound in sparser capacity settings, but may introduce computational overhead in dense-topology scenarios due to the $\mathcal O(n_s^2n_r)$ scaling of constraint additions.

6. Implications and Future Directions

The polyhedral analysis of the undirected VNE flow formulation demonstrates that the newly introduced flow-departure and flow-continuity inequalities, together with the canonical flow constraints, suffice to characterize the embedding polytope for the case of a single virtual edge routed across a substrate path. This polyhedral tightness yields direct computational benefits through stronger relaxations and reduced branch-and-bound complexity.

Ongoing research directions include extending these results to more general substrate topologies, particularly trees and larger virtual subgraphs such as stars. This suggests that further cut families or alternative description strategies may be necessary for broader classes, potentially building on the insights from the path and single-edge framework. The experimental evidence highlights the value of polyhedral analysis as a tool for improving large-scale MIP-based approaches in network embedding applications (Benhamiche et al., 16 Jan 2026).