Better Orders for Saturated Cost Partitioning in Optimal Classical Planning

Cost partitioning is a general method for adding multiple heuristic values admissibly. In the setting of optimal classical planning, saturated cost partitioning has recently been shown to be the cost partitioning algorithm of choice for pattern database heuristics found by hill climbing, systematic pattern database heuristics and Cartesian abstraction heuristics. To evaluate the synergy of the three heuristic types, we compute the saturated cost partitioning over the combined sets of heuristics and observe that the resulting heuristic is outperformed by the heuristic that simply maximizes over the three saturated cost partitioning heuristics computed separately for each heuristic type. Our new algorithm for choosing the orders in which saturated cost partitioning considers the heuristics allows us to compute heuristics outperforming not only the maximizing heuristic but even state-of-the-art planners. Introduction A∗-search (Hart, Nilsson, and Raphael 1968) with an admissible heuristic (Pearl 1984) is one of the most efficient methods for solving state-space search problems optimally. Since a single heuristic is often insufficient for challenging problems, it is usually desirable to combine the estimates of multiple heuristics admissibly. One way of doing so is to use their maximum, but this merely selects the best heuristic for each evaluated state. Cost partitioning (Katz and Domshlak 2008; Yang et al. 2008) is a general method for actually combining multiple heuristics. By dividing the original operator costs among the heuristics, it allows to sum the heuristic estimates admissibly. The resulting cost-partitioned heuristic is often much stronger than the maximum over the heuristics. In the setting of optimal classical planning (Ghallab, Nau, and Traverso 2004), it has been shown that an optimal cost partitioning can be computed in polynomial time for abstraction (Katz and Domshlak 2008; 2010) and landmark (Karpas and Domshlak 2009) heuristics. In practice, however, computing an optimal cost partitioning is often prohibitively expensive, even for a single state (Pommerening, Röger, and Helmert 2013). There are many algorithms for computing non-optimal cost partitionings (e.g., Haslum, Bonet, and Geffner 2005; Haslum et al. 2007; Katz and Domshlak 2008; Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Pommerening, Röger, and Helmert 2013). A recent addition is saturated cost partitioning (Seipp and Helmert 2014), which operates on an ordered sequence of heuristics and iteratively assigns the minimum costs that one heuristic needs for its estimates, before using the remaining costs for subsequent heuristics. Seipp, Keller, and Helmert (2017a) compared multiple cost partitioning algorithms theoretically and experimentally. They computed cost partitionings for pattern databases (PDBs) found by hill climbing (Haslum et al. 2007), systematic PDBs (Pommerening, Röger, and Helmert 2013), Cartesian abstractions (Seipp and Helmert 2013) and landmark heuristics (Karpas and Domshlak 2009), and showed that saturated cost partitioning is usually the method of choice for all tested heuristic types on the IPC benchmark collection. Their results also show that the different heuristic types have their strengths in different benchmark domains. This suggests that computing saturated cost partitioning heuristics for the combined sets of heuristics could yield an even more accurate heuristic. Testing this hypothesis is our first contribution. It turns out that combining the different heuristic types in a single cost-partitioned heuristic indeed yields a very strong heuristic. We observe that a simple maximization over three separately computed saturated cost partitioning heuristics solves even more tasks, which motivates our second contribution: by introducing a new algorithm for choosing the orders in which saturated cost partitioning considers the heuristics, we are able to compute heuristics that not only close the gap to the simple maximization heuristic, but also outperform the previous state of the art in optimal classical planning.