Hard to Solve Instances of the Euclidean Traveling Salesman Problem

The well known 4/3 conjecture states that the integrality ratio of the subtour LP is at most 4/3 for metric Traveling Salesman instances. We present a family of Euclidean Traveling Salesman instances for which we prove that the integrality ratio of the subtour LP converges to 4/3. These instances (using the rounded Euclidean norm) turn out to be hard to solve exactly with Concorde, the fastest existing exact TSP solver. For a 200 vertex instance from our family of Euclidean Traveling Salesman instances Concorde needs several days of CPU time. This is more than 1,000,000 times the runtime for a TSPLIB instance of similar size. Thus our new family of Euclidean Traveling Salesman instances may serve as new benchmark instances for TSP algorithms.