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Consider the following heuristic for building an approximate traveling - salesman tour ( cycle ) , assuming that the edge weights satisfy the triangle inequality.
Consider the following heuristic for building an approximate travelingsalesman tour cycle assuming that the edge weights satisfy the triangle inequality. Begin with a trivial cycle consisting of a singe arbitrarily chosen vertex. At each step, identify the vertex u that is not yet on the cycle but whose distance to any vertex on the cycle is minimum that is if C is the current cycle, for every vertex w C compute the number dw minv in C distw v and choose the vertex u for which du is minimum Suppose that the vertex on the cycle that is nearest to u is vertex v Extend the cycle to include u by inserting u just after v Repeat until all vertices are on the cycle.
Prove that this heuristic returns a tour whose total cost is not more than twice the cost of an optimal tour.
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