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On the Virtual Network Embedding polytope

Amal Benhamiche, Pierre Fouilhoux, Lucas Létocart, Nancy Perrot, Alexis Schneider

TL;DR

This work examines the Virtual Network Embedding (VNE) problem through a polyhedral lens, proposing new valid inequalities for the standard Flow Formulation ($FF$) to strengthen its linear relaxation. It proves that, for the special case of embedding a single virtual edge on a substrate path, the augmented flow formulation with flow-departure ($FD$) and flow-continuity ($FC$) inequalities exactly describes the VNE polytope $\\mathcal{F}(\\mathcal{G}_r,\\mathcal{G}_s,d,c)$, via a constructive flow-decomposition argument. The authors introduce ComputeFlowDecomposition to decompose any fractional solution into valid path components with no cycles, ensuring integrality of extreme points in this setting. Preliminary experiments on real and synthetic topologies show notable speedups for MIP solvers, and the authors discuss extending the results to larger subgraphs and substrate trees.

Abstract

We initiate the polyhedral study of the Virtual Network Embedding (VNE) problem, which arises in modern telecommunication networks. We propose new valid inequalities for the so-called flow formulation. We then prove, through a dedicated flow decomposition algorithm, that these inequalities characterize the VNE polytope in the case of an embedding of a virtual edge on a substrate path. Preliminary experiments show that the new inequalities propose promising speedups for MIP solvers.

On the Virtual Network Embedding polytope

TL;DR

This work examines the Virtual Network Embedding (VNE) problem through a polyhedral lens, proposing new valid inequalities for the standard Flow Formulation () to strengthen its linear relaxation. It proves that, for the special case of embedding a single virtual edge on a substrate path, the augmented flow formulation with flow-departure () and flow-continuity () inequalities exactly describes the VNE polytope , via a constructive flow-decomposition argument. The authors introduce ComputeFlowDecomposition to decompose any fractional solution into valid path components with no cycles, ensuring integrality of extreme points in this setting. Preliminary experiments on real and synthetic topologies show notable speedups for MIP solvers, and the authors discuss extending the results to larger subgraphs and substrate trees.

Abstract

We initiate the polyhedral study of the Virtual Network Embedding (VNE) problem, which arises in modern telecommunication networks. We propose new valid inequalities for the so-called flow formulation. We then prove, through a dedicated flow decomposition algorithm, that these inequalities characterize the VNE polytope in the case of an embedding of a virtual edge on a substrate path. Preliminary experiments show that the new inequalities propose promising speedups for MIP solvers.
Paper Structure (7 sections, 5 theorems, 8 equations, 4 figures, 1 algorithm)

This paper contains 7 sections, 5 theorems, 8 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Flow Formulation and VNE polytope
  3. The undirected Flow Formulation
  4. The VNE polytope
  5. Valid inequalities and equalities
  6. Case of a single virtual edge
  7. Characterization of the polytope for a substrate path

Key Result

proposition thmcounterproposition

For any $\overline{e} \in E_r$, for any $u \in V_s$, the flow-departure inequality is valid:

Figures (4)

  • Figure 1: Example of a mapping of a virtual network (on the left) on a substrate network (on the right).
  • Figure 2: Example of the augmented network
  • Figure 3: Example of a flow decomposition
  • Figure 4: Steps of the algorithm on a 4-node substrate path. On the left, the residual fractional variables; on the right, the valid paths $P$ constructed by the algorithm, a) before the algorithm, b) after the first stage on $u_1$, c) after the first stage on $u_2$, d) after the second stage on $u_3$ (algorithm is completed).

Theorems & Definitions (8)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof