I'm trying to build a game where a person needs to start from a point 1 and clear his path to point 2. I researched this and found that the A* algorithm will do fine here. The problem with it comes is that I need to have multiple targets and find the smallest number of nodes to be cleared in order to connect all of the targets. Below is a simple grid to represent what I mean with 'o' being the uncleared fields, 'x' the unclearable fields, '1-4' are the points I need to connect and '.' are the cleared fields. Considering you can go back and use already cleared fields, the total count of fields to be cleared is 13.

I checked out the topic on A* algorithm with multiple goals, but I'm not sure that it can work for me, especially in the case that some of the fields can be cleared before the algorithm starts, which will change the final result. If someone has done something similar or can just point me in the right direction.

Here the fields needed to be cleared changes to 11 and also the path between 1-2. The whole thing is a project for my university studies and the part with the multiple points is not mandatory, but I want to solve it.

Ok,
So you want me to calculate A* distance between all the points and include them into a graph and calculate the shortest path. The graph will look something like that (haven't included the weights)

And now calculate the shortest path?

A path calculation with multiple goals is normally for having to pick one of the goals afaik, while you don't want a selection, you want all goals. Maybe A* is not what you need for solving the more complicated puzzle????

That is, in your search, did you look for "smallest path between 2 points" or for "smallest path between N points with re-use of path"? That can make a large difference.

I actually did find the "Travelling Salesman problem", but didn't implement any solutions. I think that http://stackoverflow.com/questions/2501732/algorithm-shortest-path-between-all-points has the solution in the top answer, but considering I've only 4 nodes, I will try the A* algorithm first with the graph after that.

I agree with above: All pairs -> A* -> path graph -> find minimum spanning tree of path graph -> solution.

I agree with above: All pairs -> A* -> path graph -> find minimum spanning tree of path graph -> solution.

Not exactly the MST. For an optimal solution it will not always suffice considering that 2 nodes must have a degree of 1 and the other 2 must have a degree of 2. For this subset of spanning trees, we want to find the minimal one.

(Sorry for nitpicking :P)