Iterative Refinement of a Solution

Many exponentially-hard problems can be solved by searching through a space of states to determine a sequence of steps constituting a solution. Algorithms that produce optimal solutions (e.g., shortest path) generally require greater computational resources (e.g., time) than their sub-optimal counterparts. Consequently, many optimal algorithms cannot produce any usable solution when the amount of time available is limited or hard to predict in advance. Anytime algorithms address this problem by initially finding a suboptimal solution very quickly and then generating incrementally better solutions with additional time, effectively providing the best solution generated so far anytime it is required. In this research, we generate initial solutions cheaply using a fast search algorithm. We then improve this low-quality solution by identifying subsequences of steps that appear, based on heuristic estimates, to be considerably longer than necessary. Finally, we perform a more expensive search between the endpoints of each subsequence to find a shorter connecting path. We will show that this improves the overall solution incrementally over time while always having a valid solution to return whenever time runs out. We present results that demonstrate in several problem domains that AIRS (Anytime Iterative Refinement of a Solution) rivals other widelyused and recognized anytime algorithms and also produces results comparable to other popular (but not anytime) heuristic algorithms such as Bidirectional A* search. Motivation: Greedy Plateaus Inexpensive searches can be used to generate low-quality solutions quickly. In particular, we begin by using best-first greedy search (“Greedy”)—in which the search is guided solely by the heuristic estimated distance to the goal (denoted h)—to generate an initial low-quality solution. In domains in which “low-quality” implies “longer” (e.g., more actions), these long solutions often contain one or more greedy plateaus. A greedy plateau is comprised of a sequence of states that all remain at approximately the same estimated distance (h value) from the goal. This apparent “orbit” of the goal can often make up the majority of the solution. Copyright c © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Figure 1: This graph shows a low-quality (270-step) solution with three greedy plateaus. For each step on the x-axis, the estimated distance from that state to the goal (h value) is plotted. Figure 1 provides a Greedy Solution in which we observe three such plateaus: one from steps 50-80, another from 90130, and the third from 150-250. Greedy plateaus often result from greedy searches where h, while admissible, badly underestimates the actual remaining distance to the goal. After most of the states on the “orbit” are visited, h values eventually improve only to settle on other plateaus later, as shown above. The motivation for the algorithm we present, called AIRS, is based on the observation that most plateaus should be fairly easy to identify and to shorten with a better, more memory-intensive search (e.g., Bidirectional A*). We also observe that the larger the number of states on a plateau (with the same h value), the greater is the probability that pairs of states near the extremes of the plateau will have a much shorter path between them than is reflected in the greedy solution. The unnecessarily longer path can then be replaced with the short-cut, eliminating the wasteful segment. This is the crux of iterative refinement as embodied in the AIRS algorithm. 26 Proceedings of the Twenty-Fifth International Florida Artificial Intelligence Research Society Conference