Understanding an Inconsistent Heuristic

Question

Say I have a grid with some squares designated as "goal" squares. I am using A* in order to navigate this grid, trying to visit every goal square at least once using non-diagonal movement. Once a goal square has been visited, it is no longer considered a goal square. Think Pac Man, moving around and trying to eat all the dots.

I am looking for a consistent heuristic to give A* to aid in navigation. I decided to try a "return the Manhattan Distance to the nearest unvisited goal" heuristic for any given location. I have been told that this is not a consistent heuristic but I do not understand why.

Moving one square towards the closest goal square has a cost of one, and the Manhattan Distance should also be reduced by one. Landing on a goal square will either increase the value of the heuristic (because it will now seek the next nearest unvisited goal) or end the search (if the goal was the last unvisited goal)

H(N) < c(N,P) + h(P) seems to always hold true. What is it that makes this algorithm inconsistent, or is my instructor mistaken?

Answer 1

If you are asking how to use A* to find the shortest path through all the goals, the answer is: you can't (with only one iteration). This is the Travelling Salesman Problem, an NP-Complete problem. To solve this using A*, you'd need to try every permutation of goal-orderings. Each path from a single-start to a single-goal could then be solved using A* (so you'd need to run the algorithm multiple times for each permutation).

However, if you are asking how to use A* to find the shortest path from a single start to any one of a number of goals, your solution works fine, and your heuristic is indeed consistent. The minimum of multiple consistent-heuristics is still a consistent-heuristic, which is easy to prove.