M. Tieves

On the computational complexity of the virtual network embedding problem

Given a graph representing a substrate (or physical) network with node and edge capacities and a set of virtual networks with node capacity demands and node-to-node traffic demands, the Virtual Network Embedding problem (VNE) calls for an embedding of (a subset of) the virtual networks onto the substrate network which maximizes the total profit while respecting the physical node and edge capacities. In this work, we investigate the computational complexity of VNE. In particular, we present a polynomial-time reduction from the maximum stable set problem which implies strong NP-hardness for VNE even for very special subclasses of graphs and yields a strong inapproximability result for general graphs. We also consider the special cases obtained when fixing one of the dimensions of the problem to one. We show that VNE is still strongly NP-hard when a single virtual network request is present or when each virtual network request consists of a single virtual node and that it is weakly NP-hard for the case with a single physical node.

Robust Redundancy Elimination for Energy-Aware Routing

Many studies have shown that energy-aware routing (EAR) can significantly reduce energy consumption of a backbone network. Redundancy Elimination (RE) techniques provide a complementary approach to reduce the amount of traffic in the network. In particular, the GreenRE model combines both techniques, offering potentially significant energy savings. We propose a concept for respecting uncertain rates of redundant traffic within the GreenRE model, closing the gap between theoretical modeling and drawn-from life data. To model redundancy rate uncertainty, the robust optimization approach of Bertsimas and Sim (2004) is adapted and the problem is formally defined as mixed integer linear program. An exemplary evaluation of this concept with real-life traffic traces and estimated fluctuations of data redundancy shows that this closer-to-reality model potentially offers significant energy savings in comparison to GreenRE and EAR.

Virtual network embedding under uncertainty: Exact and heuristic approaches

Given a physical substrate network and a collection of requests of virtual networks, the Virtual Network Embedding problem (VNE) calls for the embedding onto the physical substrate of a selection of virtual networks in such a way that the profit is maximized. The embedding corresponds to a virtual-to-physical mapping of nodes and links, subject to capacity constraints. Since, in practical scenarios, node and link demands are typically much smaller than the peak values specified in the virtual network requests, in this work we propose and investigate a robust optimization approach. This allows us to find solutions with a much larger profit which, at the same time, are guaranteed to be feasible with a high probability. To this end, we propose a robust Mixed-Integer Linear Programming (MILP) formulation for VNE, based on the well-known model of Γ-robustness. To solve larger scale instances, for which the exact approach is computationally too demanding, we also propose a MILP-based two-phase heuristic which relies on Γ-robustness.

Extended Cutset Inequalities for the Network Power Consumption Problem

Abstract In this paper, we enhance the MIP formulation for the Network Power Consumption problem, proposed by Giroire et al. We derive cutting planes, extending the wellknown cutset inequalities, and report on preliminary computations.

Data Uncertainty in Virtual Network Embedding: Robust Optimization and Protection Levels

We address the virtual network embedding problem (VNE) which, given a physical (substrate) network and a collection of virtual networks (VNs), calls for an embedding of the most profitable subset of VNs onto the physical substrate, subject to capacity constraints. In practical applications, node and link demands of the different VNs are, typically, uncertain and difficult to know a priori. To face this issue, we first model VNE as a chance-constrained Mixed-Integer Linear Program (MILP) where the uncertain demands are assumed to be random variables. We then propose a

Column Generation for Frequency Assignment in Slow Frequency Hopping Networks

\varGamma

Discrete and robust optimization approaches to network design with compression and virtual network embedding

Γ-robust optimization approach to approximate the original chance-constrained formulation, capable of yielding solutions with a large profit that are feasible for almost all the possible realizations of the uncertain demands. To solve larger scale instances, for which the exact approach is computationally too demanding, we propose two MILP-based heuristics: a parametric one, which relies on a parameter setting chosen a priori, and an adaptive one, which does not. We conclude by reporting on extensive computational experiments where the different methods and approaches are compared.

Robust spectrum allocation in elastic flexgrid optical networks: Complexity and formulations

We propose a new cutting plane algorithm for Integer Linear Programming, which we refer to as the bound-optimal cutting plane method. The algorithm amounts to simultaneously generating k cuts which, when added to the linear programming relaxation, yield the provably largest bound improvement. We show that, in the general case, the corresponding cut generating problem can be cast as a Quadratically Constrained Quadratic Program. We also show that, for a large family of cuts, the latter can be reformulated as a Mixed-Integer Linear Program. We present computational experiments on the generation of bound-optimal stable set and cover inequalities for the max clique and knapsack problems. They show that, with respect to standard algorithms, the bound-optimal cutting plane method allows for a substantial reduction in the number of cuts and iterations needed to achieve either a given bound or an optimal solution.