In this talk, we focus on regions that are hyperplanes in the $d$-dimensional Euclidean space. While for $d=2$ an exact algorithm with running time $O(n^5)$ is known, settling the exact approximability of the problem for $d=3$ has been repeatedly posed as an open question. To date, only an approximation algorithm with guarantee exponential in $d$ is known, and NP-hardness remains open.
For arbitrary fixed $d$, we develop a Polynomial Time Approximation Scheme (PTAS) that works for both the tour and path version of the problem. Our algorithm is based on approximating the convex hull of the optimal tour by a convex polytope of bounded complexity. Such polytopes are represented as solutions of a sophisticated LP formulation, which we combine with the enumeration of crucial properties of the tour. As the approximation guarantee approaches~$1$, our scheme adjusts the complexity of the considered polytopes accordingly.
In the analysis of our approximation scheme, we show that our search space includes a sufficiently good approximation of the optimum. To do so, we develop a novel sparsification technique to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the above sense. Hereby, we maintain important properties of the polytope.
We believe that both the LP formulation representing polytopes of adjustable complexity and the sparsification techniques are of independent interest.