On the complexity of approximating Euclidean traveling salesman tours and minimum spanning trees
We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in ℝd, where d≥ 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be θ(nlog(n/r)), for all n and r such that r<n 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=θ(n 1−1/d ).
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