Stefano Coniglio

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.

Network Optimization Problems Subject to Max-Min Fair Flow Allocation

We propose a novel way to consider the max-min fairness (MMF) paradigm in traffic engineering. Since MMF appears as a reference model for a fair capacity allocation when the traffic flows are elastic and rates are adapted based on resource availability, we consider it as a requirement due to the way resources are shared by the distributed rate control scheme (like that of the transport protocol), rather than the routing objective. In particular, we define the traffic engineering problem where, given a network topology with link capacities and a set of elastic traffic demands to route, we must select a single path for each demand so as to maximize a network utility function, assuming an MMF bandwidth allocation. We propose a compact mixed-integer linear programming formulation as well as a restricted path formulation. We show with computational experiments that the exact formulation can be solved in a reasonable amount of computing time for medium-size networks and that the restricted path model provides solutions of comparable quality much faster.

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.

Coordinated cutting plane generation via multi-objective separation

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.

On single-path network routing subject to max-min fair flow allocation

Abstract Fair allocation of flows in multicommodity networks has been attracting a growing attention. In Max-Min Fair (MMF) flow allocation, not only the flow of the commodity with the smallest allocation is maximized but also, in turn, the second smallest, the third smallest, and so on. Since the MMF paradigm allows to approximate the TCP flow allocation when the routing paths are given and the flows are elastic, we address the network routing problem where, given a graph with arc capacities and a set of origin-destination pairs with unknown demands, we must route each commodity over a single path so as to maximize the throughput, subject to the constraint that the flows are allocated according to the MMF principle. After discussing two properties of the problem, we describe a column generation based heuristic and report some computational results.

Energy-aware traffic engineering with elastic demands and MMF bandwidth allocation

In recent years, there has been a remarkable growth of the Internet energy consumption, which is expected to persist in the future at an even higher pace. At the same time the network access capacity of individual subscribers is rapidly reaching values high enough to move the traffic bottleneck from the access network to the core network in most scenarios. This will soon make the elastic nature of traffic an important aspect of network resource management and will require a redesign of the energy-aware traffic engineering techniques so far based on inelastic traffic demands. We propose a novel optimization approach to select a routing path for each elastic traffic demand and decide which routers and links to put to sleep so as to maximize a network utility measure depending on the traffic demand rates, while satisfying a constraint on the total energy consumption. Bandwidth is allocated to each elastic demand according to the Max-Min Fairness (MMF) paradigm, which approximates the resource allocation of the transport layer.

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

On the Generation of Cutting Planes which Maximize the Bound Improvement

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