foundations of computational agents
The third edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2023 is now available (including full text).
An admissible heuristic is a non-negative function $h$ of nodes, where $h(n)$ is never greater than the actual cost of the shortest path from node $n$ to a goal. The standard way to construct a heuristic function is to find a solution to a simpler problem, which is one with fewer constraints. A problem with fewer constraints is often easier to solve (and sometimes trivial to solve). An optimal solution to the simpler problem cannot have a higher cost than an optimal solution to the full problem because any solution to the full problem is a solution to the simpler problem.
In many spatial problems where the cost is distance and the solution is constrained to go via predefined arcs (e.g., road segments), the straight-line Euclidean distance between two nodes is an admissible heuristic because it is the solution to the simpler problem where the agent is not constrained to go via the arcs.
For many problems one can design a better heuristic function, as in the following examples.
Consider the delivery robot of Example 3.3, where the state space includes the parcels to be delivered. Suppose the cost function is the total distance traveled by the robot to deliver all the parcels. If the robot could carry multiple parcels, one possible heuristic function is the maximum of (a) and (b):
the maximum delivery distance for any of the parcels that are not at their destination and not being carried, where the delivery distance of a parcel is the distance to that parcel’s location plus the distance from that parcel’s location to its destination
the distance to the furthest destination for the parcels being carried.
This is not an overestimate because it is a solution to the simpler problem which is to ignore that it cannot travel though walls, and to ignore all but the most difficult parcel. Note that a maximum is appropriate here because the agent has to both deliver the parcels it is carrying and go to the parcels it is not carrying and deliver them to their destinations.
If the robot could only carry one parcel, one possible heuristic function is the sum of the distances that the parcels must be carried plus the distance to the closest parcel. Note that the reference to the closest parcel does not imply that the robot will deliver the closest parcel first, but is needed to guarantee that the heuristic is admissible.
In route planning of Example 3.1, when minimizing time, a heuristic could use the straight-line distance from the current location to the goal divided by the maximum speed – assuming the user could drive straight to the destination at top speed.
A more sophisticated heuristic may take into account the different maximum speeds on highways and local roads. One admissible heuristic is the minimum of (a) and (b):
the estimated minimum time required to drive straight to the destination on slower local roads
the minimum time required to drive to a highway on slow roads, then drive on highways to a location close to the destination, then drive on local roads to the destination.
The minimum is appropriate here because the agent can go via highways or local roads, whichever is quicker.
In the above examples, determining the heuristic did not involve search. Once the problem is simplified, it could be solved using search, which should be simpler than the original problem. Note the simpler search problem needs to be solved multiple times, even perhaps for all nodes. It is often useful to cache these results into a pattern database that maps the nodes of the simpler problem into the heuristic value. In the simpler problem, there are often fewer nodes, and so multiple original nodes are mapped into a single simpler node, so this may be feasible.