Euclidean TSP in Narrow Strips - Alkema, Henk; de Berg, Mark; van der Hofstad, Remco; Kisfaludi-Bak, Sándor (Discrete & computational geometry, 2024)
We investigate how the complexity of Euclidean TSP for point sets P inside the strip ( - ∞ , + ∞ ) × [ 0 , δ ] depends on the strip width δ . We obtain two main results. For the case where the points have distinct integer x -coordinates, we prove that a shortest bitonic tour (which can be computed in O ( n log 2 n ) time using an existing algorithm) is guaranteed to be a shortest tour overall when δ ⩽ 2 2 , a bound which is best possible. We present an algorithm that is fixed-parameter tractable with respect to δ . Our algorithm has running time 2 O ( δ ) n + O ( δ 2 n 2 ) for sparse point sets, where each 1 × δ rectangle inside the strip contains O (1) points. For random point sets, where the points are chosen uniformly at random from the rectangle [ 0 , n ] × [ 0 , δ ] , it has an expected running time of 2 O ( δ ) n . These results generalise to point sets P inside a hypercylinder of width δ . In this case, the factors 2 O ( δ ) become 2 O ( δ 1 - 1 / d ) .