What is known about the LP gap of (the natural Held-Karp relaxation of) the (symmetric) Travelling salesman in special instances?
I'm only aware of one special case where the extreme points are all half integral and a 7/5 in the graphic case. Is anything better than 3/2 known for say planar, euclidean, bounded treewidth/branchwidth or other easy instances?
What are the simplest instances for which the LP gap is not yet known? References/surveys appreciated.