Euclidean TSP in Narrow Strips

Abstract

We investigate how the complexity of Euclidean TSP for point sets P inside the strip \((-\infty ,+\infty )\times [0,\delta ]\) depends on the strip width \(\delta \). 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 \(\delta \leqslant 2\sqrt{2}\), a bound which is best possible.

• We present an algorithm that is fixed-parameter tractable with respect to \(\delta \). Our algorithm has running time \(2^{O(\sqrt{\delta })} n + O(\delta ^2 n^2)\) for sparse point sets, where each \(1\times \delta \) rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle \([0,n]\times [0,\delta ]\), it has an expected running time of \(2^{O(\sqrt{\delta })} n\). These results generalise to point sets P inside a hypercylinder of width \(\delta \). In this case, the factors \(2^{O(\sqrt{\delta })}\) become \(2^{O(\delta ^{1-1/d})}\).