Automatically generated lower bounds for search

Abstract

Heuristic search algorithms (eg. A* and IDA*) with accurate lower bounds can solve impressively large problems optimally. Most lower bounds, such as the well known Manhattan Distance heuristic for the sliding-tile puzzles or the Assignment Problem lower bound for the Asymmetric Traveling Salesman problem, are the products of human ingenuity and insight. An alternative approach to obtain lower bounds is to precalculate shortest distances in an abstraction of the original search space which is derived automatically and store the bounds in pattern databases (look-up tables). This latter technique, based on the ideas of Culberson and Schaeffer, gained popularity when Korf for the first time solved random instances of Rubik's Cube using pattern databases. While researchers were pushing for solving larger and larger problems, the fact that there exist a very large number of abstract spaces that can provide lower bounds was overlooked. This thesis fills this gap in research by investigating the search performance of lower bounds derived from abstractions. We also use the results of this analysis to automatically derive high performance pattern databases. First, we establish a very predictable trade-off between search speed and the number of entries in the pattern database. Second, we derive simple statistics that can predict the search performance of pattern databases without performing actual searches in the original state space. Using these results, we derive high performance pattern databases to search for macro-operators and to solve challenging instances of the well known Sequential Ordering Problem (SOP). Macro-search is a good candidate to showcase automatically derived lower bounds since there are many search spaces and each needs a different lower bound. The SOP is an NP-hard optimization problem. We were able to solve an unsolved instance from the TSPLIB. This required a greedy search in the space of abstractions to find a sufficiently accurate lower bound and several novel enhancements to the basic branch and bound algorithm.