Travelling salesman (with predefined edges) heuristics?

Question

I'm looking for an algorithm that is faster than exponential which will find ANY cycle in a traveling salesman problem. It doesn't matter how bad the cycle is, it just needs to be a cycle. What I'm really looking for, then, is an algorithm for a hamiltonian circuit. Something that will start at a point, reach all other points, and then end at the starting point on a graph like this: http://neogen.amdusers.com/wikipics/projects/tsp.png

So far I have found this random algorithm which did not seem to work for my example case: http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Hamiltonian_path_problem.html

And "Palmer's algorithm" which I'm having trouble understanding: Palmer's Algorithm for Hamiltonian cycles

Are there more than these 2 algorithms for doing this?