consider the following closest-point heuristic for building an approximate traveling- salesperson tour whose cost function satisfies the triangle inequality. begin with a trivial cycle consisting of a single arbitrarily chosen vertex. at each step, identify the vertex u that is not on the cycle but whose distance to any vertex on the cycle is minimum. suppose that the vertex on the cycle that is nearest 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.