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 − 1)h ≤ h ≤ (1 + 2)h, where 1, 2 ∈ [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 1 + 2 < 1 and the number of near-optimal solutions in the search tree is not too large. We go on to 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 nearoptimal 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.