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We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in d, where d 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be (nlog(n/r)), for all n and r such that r and r is larger than some constant. In the more powerful model of computation that additionally uses the floor function and random access, both problems can be solved in O(n) time if r=(n1–1/d).

On the Complexity of Approximating Euclidean Traveling Salesman Tours and Minimum Spanning Trees