Euclidean TSP in Narrow Strips

We investigate how the complexity of Euclidean TSP for point sets P inside the strip (-infinity,+infinity)x[0,delta] depends on the strip width delta . - We obtain two main results.For the case where the points have distinct integer -coordinates, we prove that a shortest bitonic tour (which can be computed in O(nlog(2)n) time using an existing algorithm) is guaranteed to be a shortest tour overall when delta <= 2 root 2 , a bound which is best possible. - We present an algorithm that is fixed-parameter tractable with respect to delta . Our algorithm has running time 2(O(root delta)n)+O(delta(2)n(2)) for sparse point sets, where each 1x delta rectangle inside the strip contains (1) points. For random point sets, where the points are chosen uniformly at random from the rectangle [0,n]x[0,delta] , it has an expected running time of 2(O(root delta)n) . These results generalise to point sets inside a hypercylinder of width delta . In this case, the factors 2(O(root delta)) become 2(O(delta 1-1/d)) .