Geometric TSP on sets

In One-of-a-Set TSP, also known as the Generalised TSP, the input is a collection $P:=\{P_1,...,P_r\}$ of sets in a metric space and the goal is to compute a minimum-length tour that visits one element from each set.

In the Euclidean variant of this problem, each $P_i$ is a set of points in $R^d$. Let $H_i$ be a hypercube that contains $P_i$, for $1⩽i⩽r$. We investigate how the complexity of Euclidean One-of-a-Set TSP depends on λ, the ply of the set $H:=\{H_1,...,H_r\}$ of hypercubes. (The ply is the smallest λ such that every point in $R^d$ is contained in at most λ of the hypercubes). We show that the problem can be solved in $2^{O\left({\lambda }^{1/d}{n}^{1-1/d}\right)}$ time, where $n:={\sum }_{i=1}^r\left|P_i\right|$ is the total number of points, and that the problem cannot be solved in $2^{o\left(n\right)}$ time when $\lambda =\Theta \left(n\right)$, unless the Exponential Time Hypothesis (ETH) fails.

In Rectilinear One-of-a-Cube TSP, the input is a set $H$ of hypercubes in $R^d$ and the goal is to compute a minimum-length rectilinear tour that visits every hypercube. We show that the problem can be solved in $2^{O({\lambda }^{1/d}{n}^{1-1/d}\log n)}$ time, where n is the number of hypercubes.