NOTE: Due to the fact that the trip does not end at the same place it started and also the fact that every point can be visited more than once as long as I still visit all of them, this is not really a TSP variant, but I put it due to lack of a better definition of the problem.
So..
Suppose I am going on a hiking trip with n points of interest. These points are all connected by hiking trails. I have a map showing all trails with their distances, giving me a directed graph.
My problem is how to approximate a tour that starts at a point A and visits all n points of interest, while ending the tour anywhere but the point where I started and I want the tour to be as short as possible.
Due to the nature of hiking, I figured this would sadly not be a symmetric problem (or can I convert my asymmetric graph to a symmetric one?), since going from high to low altitude is obviously easier than the other way around.
Also I believe it has to be an algorithm that works for non-metric graphs, where the triangle inequality is not satisfied, since going from a to b to c might be faster than taking a really long and weird road that goes from a to c directly. I did consider if triangle inequality still holds, since there are no restrictions regarding how many times I visit each point, as long as I visit all of them, meaning I would always choose the shortest of two distinct paths from a to c and thus never takr the long and weird road.
I believe my problem is easier than TSP, so those algorithms do not fit this problem. I thought about using a minimum spanning tree, but I have a hard time convincing myself that they can be applied to a non-metric asymmetric directed graph.
What I really want are some pointers as to how I can come up with an approximation algorithm that will find a near optimal tour through all n points
You can simplify this problem to a normal TSP problem with n+1 vertexes. To do this, take node 'A' and all the points of interest and compute a shortest path between each pair of these points. You can use the all-pairs shortest path algorithm on the original graph. Or, if n is significantly smaller than the original graph size, use single-source shortest path algorithm for these n+1 vertexes. Also you can set length of all the paths, ending at 'A', to some constant, larger than any other path, which allows to end the trip anywhere (this may be needed only for TSP algorithms, finding a round-trip path).
As a result, you get a complete graph, which is metric, but still asymmetric. All you need now is to solve a normal TSP problem on this graph. If you want to convert this asymmetric graph to a symmetric one, Wikipedia explains how to do it.