Need an idea for A star search algorithm with multiple goals

Question

The algorithm of A star search with a specified goal is pretty straightforward. However, what if there are multiple goals in a graph. For instance; you may want to find a shortest path that must include previously specified nodes. Constraint here is say your path must include A, B and C nodes( or more) not just find a path to node A or B or C. And of course the graph includes one or more A, B, C type nodes. So there is a question how can I adapt the A star search algorithm for multiple goals?

edit : we can visit nodes more than one.

Answer 1

You are describing conditions on path and not conditions on goal. A*, like all search algorithms - is finding a path to a goal [could be in a set, of goal, no problems with that].

Your problem [for the general case] is at least as hard as the Traveling Salesman Problem, and thus this problem is NP-Hard.

The reduction is simple: Given an instance of TSP - find the shortest path from a certain v to v such that the path is going through all vertices [constraint]. You can do it by simply marking each vertex with a different mark.

Note however, that A* algorithm has no problem to find shortest path to a vertex in a set of goal vertices. Remember that A* is based on Dijkstra's Algorithm, which is finding shortest path to all vertices from a single source.