Taking into account the dynamic nature of traffic in telecommunication networks, the robust network design problem is to fix the edge capacities so that all demand vectors belonging to a polytope can be routed. While a common heuristic for this co‐NP‐hard problem is to compute, in polynomial time, an optimal static routing, affine routing can be used to obtain better solutions. It consists in restricting the routing to affinely depend on the demands. We show that a node‐arc formulation is less conservative than an arc‐path formulation. We also provide a cycle‐based formulation that is equivalent to the node‐arc formulation. To further reduce the solution's cost, several new formulations are obtained by relaxing flow conservation constraints and aggregating demands. As might be expected, aggregation allows us to reduce the size of formulations. A more striking result is that aggregation reduces the solution's cost.
Considering the dynamic nature of traffic, the robust network design problem consists in computing the capacity to be reserved on each network link such that any demand vector belonging to a polyhedral set can be routed. The objective is either to minimize congestion or a linear cost. And routing freely depends on the demand.We first prove that the robust network design problem with minimum congestion cannot be approximated within any constant factor. Then, using the ETH conjecture, we get a Ω( log n log log n ) lower bound for the approximability of this problem. This implies that the well-known O(log n) approximation ratio established by Räcke in 2008 is tight.Using Lagrange relaxation, we obtain a new proof of the O(log n) approximation. An important consequence of the Lagrange-based reduction and our inapproximability results is that the robust network design problem with linear reservation cost cannot be approximated within any constant ratio. This answers a long-standing open question of Chekuri.Finally, we show that even if only two given paths are allowed for each commodity, the robust network design problem with minimum congestion or linear costs is hard to approximate within some constant k.
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