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2

How about: Build a weighted complete bipartite graph where the left vertex set is $S$, the right vertex set is $G$, and the weight of the edge between $s \in S$ and $g \in G$ is the 3-D Manhattan distance from $s$ to $g$. Solve the assignment problem for this graph. This finds the exact minimum. It does unfortunately take about $O(n^3)$ for a typical ...

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A good initial state can often be helpful. I would guess that there is a significant chance that spending extra time to find a good initial state will be useful. All we can say in general is that "it depends". It depends on the specific problem and the shape of the fitness landscape. If the problem has few or no local minima, the search might ...

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