Optimized TSP Algorithms

Question

I am interested in ways to improve or come up with algorithms that are able to solve the Travelling salesman problem for about n = 100 to 200 cities.

Th

Answer 1

I have a theory, but I've never had the time to pursue it:

The TSP is a bounding problem (single shape where all points lie on the perimeter) where the optimal solution is that solution that has the shortest perimeter.

There are plenty of simple ways to get all the points that lie on a minimum bounding perimeter (imagine a large elastic band stretched around a bunch of nails in a large board.)

My theory is that if you start pushing in on the elastic band so that the length of band increases by the same amount between adjacent points on the perimeter, and each segment remains in the shape of an eliptical arc, the stretched elastic will cross points on the optimal path before crossing points on non-optimal paths. See this page on mathopenref.com on drawing ellipses--particularly steps 5 and 6. Points on the bounding perimeter can be viewed as focal points of the ellipse (F1, F2) in the images below.

What I don't know is if the "bubble stretching" process needs to be reset after each new point is added, or if the existing "bubbles" continue to grow and each new point on the perimeter causes only the localized "bubble" to turn into two line segments. I'll leave that for you to figure out.