The Time Complexity Of A* With Approximate Heuristics On Multiple-Solution Search Spaces

We study the behavior of the A* search algorithm when coupled with a heuristic h satisfying (1 - epsilon(1))h* <= h <= (1 + epsilon(2))h* , where epsilon(1,) epsilon(2) is an element of [0, 1) are small constants and h* denotes the optimal cost to a solution. We prove a rigorous, general upper bound on the time complexity of A* search on trees that depends on both the accuracy of the heuristic and the distribution of solutions. Our upper bound is essentially tight in the worst case;in fact, we show nearly matching lower bounds that are attained even by non-adversarially chosen solution sets induced by a simple stochastic model. A consequence of our rigorous results is that the effective branching factor of the search will be reduced as long as epsilon(1) + epsilon(2) < 1 and the number of near-optimal solutions in the search tree is not too large. We go onto provide an upper bound for A* search on graphs and in this context establish a bound on running time determined by the spectrum of the graph.We then experimentally explore to what extent our rigorous upper bounds predict the behavior of A* in some natural, combinatorially-rich search spaces. We begin by applying A* to solve the knapsack problem with near-accurate admissible heuristics constructed from an efficient approximation algorithm for this problem. We additionally apply our analysis of A* search for the partial Latin square problem, where we can provide quite exact analytic bounds on the number of near optimal solutions. These results demonstrate a dramatic reduction in effective branching factor of A* when coupled with near-accurate heuristics in search spaces with suitably sparse solution sets.