This paper presents several findings on the problem of optimal routing.

Consider the problem of routing traffic on a topology. The traffic pattern is described by a demand matrix for which each element $(s,t)$ describes the demand from $s$ to $t$. Each arc in the network has a specified capacity. The load on an arc, however, depends on the routing. The paper models the problem as an optimization on the sum of costs amid satisfying the traffic demand. The cost is defined as the sum of all link costs, which is a function of the load and capacity of the link. The cost function is a piecewise linear convex function in the paper.

The first finding is that, using OSPF, multipath routing is done by even split. This is less flexible and less efficient, compared to MPLS. By a counter example, it is shown in the paper that no clever weight-setting is available for OSPF to be as optimal as MPLS. The example shows that (with proof), the optimality in OSPF is as worse as 5000 times the optimality in MPLS. Which the figure 5000 is from the cost function, defining the worse case to be as that much costly.

The second finding is that, in real networks, because the topology is not as weird as the synthetic case, the optimality in OSPF is close to the MPLS optimal, within a few percent off.

This paper presents a linear programming model for general (MPLS) routing in page 3. Adding the even split constraint, which correspond to the OSPF, makes the model nonlinear and NP-hard. To solve this problem, this paper proposed a local search heuristic. Given a weight vector (vector of all weights for all arcs in the network), it searches its neighbour for optimizing the cost function. The neighbourhood is defined as:

• Changing one single weight in the vector; or
• Balancing sum of weights of several paths a node and another

Such search is slow and inefficient because, like simulated annealing, loop may occur on the search space. To avoid repeating on the same weight vector, a hash table is used to decide if a vector is new. This is claimed to be speeding up the search process.

Another strategy to improve search efficiency is diversification. When the cost function is not improved for 300 iterations, the weight vector is randomly perturbed and the hash table is reset. This is to avoid “long valleys” in the search space, which is difficult to escape.

## Bibliographic data

@inproceedings{
title = "Internet Traffic Engineering by Optimizing OSPF Weights",
author = "Bernard Fortz and Mikkel Thorup",
booktitle = "Proc INFOCOM",
pages = "519--528",
year = "2000",
}