Not All Insertion Methods Yield Constant Approximate Tours in the Euclidean Plane

An insertion heuristic for the traveling salesman problem adds cities iteratively to an existing tour by replacing one edge with a two-edge path through the new city in the cheapest possible way. Rosenkrantz, Stearns, and Lewis asked whether every order of inserting vertices gives a constant-factor approximation algorithm. We answer this question by showing that for some point sets, there is an order that yields tours with length (logn= log logn) times optimum, even if the underlying metric space is the Euclidean plane.