Towards Robust Network Design using Integer Linear Programming Techniques Arie M. C. A. Koster RWTH Aachen University Lehrstuhl II f¨ ur Mathematik W¨ ullnerstr. 5b D-52062 Aachen, Germany Email: koster@math2.rwth-aachen.de Manuel Kutschka RWTH Aachen University Lehrstuhl II f¨ ur Mathematik W¨ ullnerstr. 5b D-52062 Aachen, Germany Email: kutschka@math2.rwth-aachen.de Christian Raack Zuse Institute Berlin (ZIB) Department Optimization Takustr. 7 D-14195 Berlin, Germany Email: raack@zib.de Abstract—Traffic in communication networks fluctuates heav- ily over time. Thus, to avoid capacity bottlenecks, operators highly overestimate the traffic volume during network planning. In this paper we consider telecommunication network design under traffic uncertainty, adapting the robust optimization ap- proach of [11]. We present three different mathematical formu- lations for this problem, provide valid inequalities, study the computational implications, and evaluate the realized robustness. To enhance the performance of the mixed-integer program- ming solver we derive robust cutset inequalities generalizing their deterministic counterparts. Instead of a single cutset inequality for every network cut, we derive multiple valid inequalities by exploiting the extra variables available in the robust formulations. For realistic networks and live traffic measurements we com- pare the formulations and report on the speed up by the valid inequalities. We study the “price of robustness” and evaluate the approach by analyzing the real network load. The results show that the robust optimization approach has the potential to support network planners better than present methods. Index Terms—network design, robust optimization, price of robustness, integer linear programming, cutset inequalities I. I NTRODUCTION Dimensioning telecommunication networks is a complex task and it is crucial for the behavior and flexibility of the resulting network. Network design typically involves decisions about the network topology, link capacities, and traffic routing. In the classical network design problem integer capacities (cor- responding to bandwidth batches) have to be installed on the network links at minimum cost such that all traffic demands can be realized by flow simultaneously without exceeding the link capacities. Assuming a given single traffic matrix, this problem has been studied extensively in the literature, see [6, 12, 13, 24, 31, 40] and the references therein. In practice, telecommunication networks are typically de- signed without the knowledge of actual traffic. In most ap- proaches each demand is estimated in the design process, e.g., by using traffic measurements or population statistics [14, 20]. To handle future changes in the traffic volume and distribution, these values (and consequently capacities) are (highly) over- estimated. Obviously, this approach leads to a wastage of net- work capacities, investments, and energy. To create and operate more resource- and cost-efficient telecommunication networks the uncertainty of future traffic demand has to be taken into account already in the strategic capacity design process. A number of different approaches have been proposed in the literature based on stochastic optimization (e.g., [17, 35, 42]) and robust optimization (e.g., [9, 21, 22, 29]). Robust optimization, first considered by Soyster [42], aims at finding solutions that are feasible for all realizations of data in a given (bounded) uncertainty set. Bertsimas and Sim [10, 11] introduced a way to describe uncertainty in linear programs which results in tractable robust counterparts preserving the linearity of the original problem. In addition, they introduce a parameter Γ to control the price of robustness, the trade-off between the degree of uncertainty taking into account and the cost of this additional feature. In telecommunication network design, there have been different approaches to incorporate uncertainty in the planning process. In multi-period (multi-hour) network design [43], an explicit set of demand matrices is given, and the network should be designed in such a way that each of the demand matrices can be routed non-simultaneously within the installed capacities. In this context Oriolo [38] introduces the concept of dominating demand matrices (i.e., D 1 dominates D 2 if every link capacity vector supporting D 1 also supports D 2 ). Instead of describing demand matrices explicitly, Ben- Ameur and Kerivin [7, 8] consider the optimized routing of demands that may vary within a given polytope. For network design problems this concept has mainly been applied using the hose model, a polyhedral demand uncertainty set which has been introduced in the context of virtual private networks (VPNs) [19]. In its symmetric version, the hose model defines upper bounds on the sum of the incoming and outgoing node traffic for all network nodes. Hence the hose polytope is defined by one inequality for every network node. The hose model has attracted a lot of attention in recent years, in particular, due to its nice algorithmic properties assuming continuous capacities (e.g., polynomial solvable cases, see [15, 23, 26]). Altin et al. [3] develop a compact integer linear program- ming model for virtual private network design with continuous capacities and single path routing using the hose model. Altin et al. [5] study the network design problem assuming splittable flow and integer capacities (also known as network