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Brief Background

One possible way to rectify the shortcoming of the aforementioned HT model is to reduce the computational burden being placed on the network. This can be done by constraining each city (in the TSP case) to be ``on'' only once, that is by enforcing the constraint , rather than relying on an energy penalty term to try to hopefully achieve it, as it is done in the HT original algorithm. This idea was first used and analyzed for the TSP case, among others, by Peterson and Sderberg [7]. Essentially, their approach is to replace the N neurons representing by a single N-dimensional Potts neuron using what is referred to as the mean field annealing (MFA) approach. This generally yields the same final network equations as the ``neuronal circuit'' approach of [33], but the exposition is a little bit clearer, as it is laid out in a statistical mechanics frameworkgif. As stated in the above mapping subsection, one need to compute all admissible configurations of the problem at hand. For example, when mapping the TSP onto the HT model, all of the configurations are admissible, whereas if each city is restricted to being visited only once, then only vertices are admissible. After taking a mean field approximation, saddlepoint equations are derived, the solutions of which pick out the dominant states of the network at the current temperature T.

This approach was not only an improvement over the Hopfield and Tank model but also gave more feasible (and valid) solutions. For a brief overview of the generic ``black box'' MFA algorithm using the Potts neurons, take a look at Figure 12.

In addition to the TSP implementation, Peterson and colleagues extended the technique and applied it to a case of high school scheduling [8,9]. We also have used the same model for class scheduling [10] with, unfortunately, not very impressive results.



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Mon Nov 18 19:45:42 EST 1996