Gap-G-ST (on the plane) k may be arbitrary large Unless P = NP, G-ST in the plane cannot be approximated to within any constant factor.
Problem Variants Variants: 2D unconnected, overlapping (G-ST & G-TSP) unconnected, pairwise-disjoint Variants: D 3 Holds for connected variants too.
Other Corollaries Small sets size: k-G-TSP in the plane k-G-ST in the Plane Watchman Tour and Watchman path problems in 3D cannot be approximated to within any constant, unless P=NP
If the two properties are joint: then Approximating G-TSP and G-ST in the plane to within is intractable. Approximating G-TSP and G-ST in dimension d within is intractable. Open Problems
Is 2 the approximation threshold for connected overlapping neighborhoods ? Is there a PTAS for connected, pairwise disjoint neighborhoods ? How about watchman tour and path in the plane ? Does any embedding in the plane cause at least a square root loss ? Does higher dimension impel an increase in complexity ?