How are the shortest-path and travelling-salesman problems given above similar? How are they different?

They are similar in the sense that both traverses a graph and tries to find out the path with minimum sum of the weights.

They are different because shortest-path problem finds a path between two points such that sum of the weights is minimized. Whereas, travelling-salesman problem finds the path covering all the points (start and end point is same) such that sum of the weights is minimized. Also, shortest-path problem is P complex and travelling-salesman is NP-complete.

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