Narrowing the Gap Between Saturated and Optimal Cost Partitioning for Classical Planning

In classical planning, cost partitioning is a method for admissibly combining a set of heuristic estimators by distributing operator costs among the heuristics. An optimal cost partitioning is often prohibitively expensive to compute. Saturated cost partitioning is an alternative that is much faster to compute and has been shown to offer high-quality heuristic guidance on Cartesian abstractions. However, its greedy nature makes it highly susceptible to the order in which the heuristics are considered. We show that searching in the space of orders leads to significantly better heuristic estimates than with previously considered orders. Moreover, using multiple orders leads to a heuristic that is significantly better informed than any single-order heuristic. In experiments with Cartesian abstractions, the resulting heuristic approximates the optimalions, the resulting heuristic approximates the optimal cost partitioning very closely. Introduction A∗ search with an admissible heuristic is one of the most prominent methods for optimal classical planning. Because a single heuristic is often unable to capture all relevant aspects of a planning task, it is desirable to combine information from multiple heuristics. One way of doing so admissibly is to maximize over multiple heuristic estimates in each state (Holte et al. 2006). However, this method does not really combine multiple heuristics but merely selects the most informative one in each state. Cost partitioning (CP) (Katz and Domshlak 2008; Yang et al. 2008) is a more sophisticated way of combining heuristics that often produces higher estimates than any single estimator can provide. By distributing the operator costs among the heuristics, cost partitioning allows to sum the heuristic estimates admissibly. An optimal cost partitioning (OCP) is usually too expensive to compute (e.g., Pommerening, Röger, and Helmert 2013), but multiple approximations with varying time vs. accuracy tradeoffs have been proposed, such as uniform CP (Katz and Domshlak 2007b), zero-one CP (e.g., Edelkamp 2006), and post-hoc optimization (Pommerening, Röger, and Helmert 2013). Recently, Seipp and Helmert (2014) introduced the saturated cost partitioning (SCP) algorithm, which iteratively computes an abstraction, determines the minimum operator Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. costs needed to preserve all abstract goal distances, and then repeats this process with the remaining operator costs. Saturated cost partitioning assigns costs greedily and is therefore susceptible to the order in which the abstractions are computed. Seipp and Helmert considered two orders based on Bonet and Geffner’s hadd heuristic (2001) as well as a random order. Their experimental evaluation was inconclusive, as none of the three orders consistently outperformed the others. In this paper, we study the problem of ordering abstractions for SCP in more depth. Our analysis reveals that just changing the order of abstractions in SCP can make the difference between a perfect distance estimate and a blind one. To find good orders, we propose a hill climbing search in the space of all orders. The optimized orders found by hill climbing significantly improve over Seipp and Helmert’s results. However, the results also indicate that it is often impossible to find a single order that provides good guidance across the state space: orders that are accurate on a given set of states often turn out to be poor in all others. Maximizing over SCP heuristics for multiple random orders allows us to use accurate heuristics for many different states and significantly improves over single-order heuristics. This approach is similar to the one by Karpas, Katz, and Markovitch (2011), who maximize over multiple precomputed OCP heuristics instead of multiple SCP heuristics. Our method has the advantage that we never have to compute an OCP, which can be prohibitively expensive even for a single computation. We show that our sets of SCP heuristics often contain heuristics that do not contribute any information during search. Similar to other work on heuristic subset selection (e.g., Lelis et al. 2016), we try to pick a subset of heuristics that complement each other well by actively searching for multiple diverse orders. Our strongest heuristic closely approximates the optimal cost partitioning and compares favorably to the state of the art for optimal classical planning.