Blum and Furst introduced the notion of a planning graph, which is a powerful data structure that encodes information about which states may be reachable . For the logic-based problem expressed in Formulation 2.4, consider performing reachability analysis. Breadth-first search can be used from the initial state to expand the state transition graph. In terms of the input representation, the resulting graph may be of exponential size in the number of stages. This gives precise reachability information and is guaranteed to find the goal state.
The idea of Blum and Furst is to construct a graph that is much smaller than the state transition graph and instead contains only partial information about reachability. The resulting planning graph is polynomial in size and can be efficiently constructed for some challenging problems. The trade-off is that the planning graph indicates states that can possibly be reached. The true reachable set is overapproximated, by eliminating many impossible states from consideration. This enables quick elimination of impossible alternatives in the search process. Planning algorithms have been developed that extract a plan from the planning graph. In the worst case, this may take exponential time, which is not surprising because the problem in Formulation 2.4 is NP-hard in general. Nevertheless, dramatic performance improvements were obtained on some well-known planning benchmarks. Another way to use the planning graph is as a source of information for developing search heuristics for a particular problem.