Question

There’s a map with points:

The green number next to each point is that point’s ID and the red number is the bonus for that point. I have to find fastest cycle that starts and ends at the point #1 and that gains at least x (15 in this case) bonus points. I can use cities several times; however, I will gain bonus points only once.
I have to do this with the backtracking algorithm, but I don’t really know where to start. I’ve stutied about it, but I can’t see the connection between this and a backtracking.

The output would look like this:

(1,3,5,2,1) (11.813 length)

Answer 1

Backtracking is a technique applied to reduce the search space of a problem. So, you have a problem, you have a space with optimal and non-optimal solutions, and you have to pick up one optimal solution.

A simple strategy, in your problem, is to generate all the possible solutions. However, this solution would traverse the entire space of solutions, and, some times, being aware that no optimal solution will be found.

That’s the main role of backtracking: you traverse the space of solutions and, when you reach a given point where you know no optimal answer will be achieved if the search continue on the same path, you can simply repent of the step taken, go back in the traversal, and select the step that comes right after the one you found to be helpless.

In your problem, since the nodes can be visited more than once, the idea is to maintain, for each vertex, a list of vertices sorted decreasingly by the distance from the vertex owner of the list.

Then, you can simply start in one of the vertices, and do the walk on the graph, vertex by vertex, always checking if the objective is still achievable, and backtracking in the solution whenever it’s noticed that no solution will be possible from a certain point.