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Consider the following heuristic for building an approximate traveling-salesman tour assumingthat the edge weights satisfy the triangle inequality. Begin with a trivial cycle consisting of asinge arbitrarily chosen vertex. At each step identify the vertex u that is not yet on the cyclebut whose distance to any vertex on the cycle is minimum (that is if C is the current cyclefor every vertex w 62 C compute the number d(w) = minv2C dist(w v) and choose the vertex ufor which d(u) is minimum). Suppose that the vertex on the cycle that is nearest to u is vertexv. Extend the cycle to include u by inserting u just after v. Repeat until all vertices are on thecycle. Prove that this heuristic returns a tour whose total cost is not more than twice the costof an optimal tour.