Rigorous bounds for fundamental heuristic search

Date of Completion

January 2010


Computer Science




Both A* search and local search are heuristic algorithms widely used for problem-solving in Artificial Intelligence and Combinatorial Optimization. Understanding their performance is a problem of both practical and theoretical importance. Our goal in this thesis is to fill significant gaps in the rigorous analyses of these two heuristic algorithms. ^ In particular, we study the behavior of the classical A* search algorithm when coupled with a heuristic that provides estimates, accurate to within small multiplicative factors, of the optimal cost to reach a solution. We prove a general, essentially tight upper bound on the time complexity of A* search on trees that depends on both the accuracy of the heuristic and the distribution of solution objective values. A consequence of our rigorous analysis indicates that the effective branching factor of the search will be reduced as long as the heuristic is sufficiently accurate 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. Finally, we apply our analysis of A* search for the partial Latin square problem, and compare the theoretically predicted running times with experimental data. These results demonstrate dramatic reduction in effective branching factor of A* when heuristic error is small. ^ For local search, we focus on understanding its limitations, which can be reduced to studying the problem of finding a local minimum of a real-valued black-box function on a graph. In 1983, Aldous gave the first strong lower bound for any randomized algorithm to determine a local minimum on hypercubes. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened Aldous's lower bound and gave an analogous quantum lower bound. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds for local search on vertex-transitive graphs, a significantly larger family of graphs. The techniques of our proof follow Aaronson's idea of proving classical results by quantum arguments. ^