Euclidean Group Traveling Salesman Problem

In the Euclidean group TSP we are given a set of points $P$ in the plane and a set of $m$ connected regions each containing at least one point of $P$. We need to find a tour of minimum length that visits at least one point in each region. This problem unifies the TSP with Neighborhoods problem and the Group Steiner Tree problem. We give a $(9.1\alpha+1)$-approximation algorithm for the disjoint problem, which considerably improves the current best ratio for TSP with disjoint $\alpha$-fat neighborhoods. We give the first $O(1)$-approximation algorithm for the problem with intersecting regions.