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Suppose that the vertices for an instance of the travelling salesman problem are points in a plan and that the cost c(u; v) is the Euclidean distance between points u and v. Show (i.e., prove) that an optimal tour never crosses itself.

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Algorithm problems are examined.

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Proof:
Suppose an optional solution of an TSP crosses itself as shown below, we can assume the solution starts from a point , then passes through points , then finally comes back to . The solution can ...

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