Consider the problem of finding a stable state in a Hopfield neural network, in the special case when all edge weights are positive. This corresponds to the Maximum-Cut Problem that we discussed earlier in the chapter: For every edge e in the graph G, the endpoints of G would prefer to have opposite states.
Now suppose the underlying graph G is connected and bipartite; the nodes can be partitioned into sets X and Y so that each edge has one end in X and the other in Y. Then there is a natural "best" configuration for the Hopfield net, in which all nodes in X have the state +1 and all nodes in Y have the state –1. This way, all edges are good, in that their ends have opposite states.
The question is: In this special case, when the best configuration is so clear, will the State-Flipping Algorithm described in the text (as long as there is an unsatisfied node, choose one and flip its state) always find this configuration? Give a proof that it will, or an example of an input instance, a starting configuration, and an execution of the State-Flipping Algorithm that terminates at a configuration in which not all edges are good.
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