How can the A* algorithm be applied to the traveling salesman problem?
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Using A* to solve Travelling Salesman Problem
I have recently learned that the A* algorithm can be applied to the travelling salesman problem. Bot how exactly do we define the start and the goal here, and how do we apply weights to nodes (what is the heuristic)?
Would someone explain to me how A* can be applied here?
Answer
A* is a derivative of Dijsktra, which I don't think can be used in this fashion. First, the TSP generally starts from any node. More importantly though, these algorithms seek to find the shortest path between two points, irrespective of the number of nodes visited. In fact, they depend on the fact that the shortest path from S to T via some node A, the path from S to A is irrelevant if it's the same cost.
The only way I see this functioning is if you generated a new graph representing nodes visited. For example, in your priority queue, you'd place the set of nodes visited and current node. This would lead to possibly examining $n2^n$ nodes (which is not coindientally the running time of the dynamic programming solution to the TSP http://www8.cs.umu.se/kurser/TDBAfl/VT06/algorithms/BOOK/BOOK2/NODE49.HTM). So far, not great, but by using A* instead of Dijsktra, you might be able to find a heuristic that runs in a reasonable amount of time.
To implement this, your starting node would be (1,{}) and your ending node would be (1,{1..n}). You would mimic the edge weights of the original graph. As for suggestions on the heuristic, you're on your own..