Flow Aggregation based Lagrangian Relaxation with Applications to Capacity Planning of IP Networks with Multiple Classes of Service

Abstract

Nonlinear multicommodity flow problem with integer constraints is an important subject with a wide variety of application contexts. Given the inherent difficulty of solving nonlinear integer programming problems and the lack of efficient systematic approaches, research has primarily been focused on heuristics, which may be inadequate in many cases.

In this dissertation, we present a novel Lagrangian Relaxation based approach to attack the nonlinear multicommodity flow problem with integer constraints. Unlike other methods based on the relaxation of integer constraints or nonlinear constraints, our approach relaxes the flow aggregation equation, or the relationship between individual flows and the variable representing the total amount of traffic in each link. The relaxation of the flow aggregation equation makes the resulting Lagrangian dual problem separable to simpler subproblems, which will not be possible if the problem is relaxed otherwise. The subgradient method is used to find the optimal Lagrangian multipliers.

The aforementioned method is subsequently applied to three closely related capacity planning problems to verify its effectiveness. We address the problems of link dimensioning and routing optimization for IP networks supporting DiffServ Expedited Forwarding (EF) and best effort (BE) traffic classes. We first study capacity planning in the context of legacy IP networks, where only non-bifurcated routing is allowed for traffic with the same source and destination. In the second problem, we assume the use of Multiprotocal Label Switching (MPLS), and therefore allow bifurcated routing. The third problem deals with the design of survivable MPLS networks, where spare capacity is required.

We investigated experimentally the solution quality and running time of this approach. The results from our experiments indicate that our method produces solutions that are within a few percent of the optimal solution, while the running time remains reasonable on practical-sized networks. This represents the first work for capacity planning of multi-class IP networks with non-linear performance constraints and discrete link capacity constraints. We also expect the solution method to be a promising approach for other non-linear multicommodity flow problems with integer constraints.