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By James F. Gimpel

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Each leaf node, then, may have multiple parents. The resulting structure is a rooted, directed, acyclic graph. All edges in the graph are directed toward the root node. Each node has a local energy based on its downstream net input, its upstream net input, and an external bias. Mathematically, Energyi = neti ai neti = ( Σ Unetij ) + Dneti + Biasi j 36 In this formulation, the net input to node i, neti , is broken into three contributions. Uneti j represents the net upstream input of node i from parent j.

A complete tour consists of following the node labels in increasing order, until you reach the last node. From the last node you travel back to the first. Because the edges are directed, it is clear that this is also the only legal tour. Intuitively, such instances of HC problems should be difficult. Only one tour exists in each instance. In addition, there are a large number of solutions that are almost complete tours scattered throughout the search space. Figure 10 illustrates the corresponding SAT GA payoff function for the HC problem of this type with 7 nodes.

In the NOT example, nodes are constrained to be different. In the AND and OR examples, nodes are constrained to be similar. One interpretation of this is that: If a connection between two nodes is positive, the constraint is satisfied if both nodes are true or both nodes are false. If a connection between two nodes is negative, the constraint is satisfied if the nodes are in opposite states. 35 In other words, a positive connection enforces the idea that two nodes are both true or both false. A negative connection enforces the idea that both nodes are not the same.

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