Subset-Saturated Transition Cost Partitioning

Cost partitioning admissibly combines the information from multiple heuristics for optimal state-space search. One of the strongest cost partitioning algorithms is saturated cost partitioning. It considers the heuristics in sequence and assigns to each heuristic the minimal fraction of the remaining costs that are needed for preserving all heuristic estimates. Saturated cost partitioning has recently been generalized in two directions: first, by allowing to use different costs for the transitions induced by the same operator, and second, by preserving the heuristic estimates for only a subset of states. In this work, we unify these two generalizations and show that the resulting subset-saturated transition cost partitioning algorithm usually yields stronger heuristics than the two generalizations by themselves. Introduction A∗ search with an admissible heuristic is one of the main techniques to solve planning tasks optimally (Hart, Nilsson, and Raphael 1968; Pearl 1984). Since a single heuristic is usually unable to capture enough information about the task, we often need to combine the information from multiple heuristics admissibly. The preferable way of doing so is cost partitioning (CP, Katz and Domshlak 2008; Keller et al. 2016). A transition cost partitioning for a weighted transition system distributes the cost of each transition over multiple heuristics such that the sum of costs for each transition does not exceed the original transition cost. If each component heuristic is admissible, then the overall cost-partitioned heuristic, i.e., the sum of the component heuristics evaluated under the partitioned cost functions, is also admissible. Many cost partitioning papers from the literature consider an important special case of transition cost partitioning: operator cost partitioning (e.g., Haslum, Bonet, and Geffner 2005; Haslum et al. 2007; Katz and Domshlak 2008, 2010; Pommerening, Röger, and Helmert 2013). In an operator cost partitioning, all transitions that are induced by the same operator have the same cost. One of the strongest approaches to compute costpartitioned heuristics is saturated cost partitioning (Seipp, Keller, and Helmert 2020). Saturated cost partitioning considers the heuristics in sequence and assigns the first heurisCopyright © 2021, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. all states subset of states operators (a) saturated operator CP (c) subset-saturated operator CP transitions (b) saturated transition CP (d) subset-saturated transition CP ge ne ra liz at io n generalization Table 1: Saturated operator cost partitioning and its generalizations from operators to transitions and from all states to a subset of states. tic the minimal fraction of the remaining costs that it needs to preserve its heuristic estimates. Then the algorithm uses the remaining costs to treat the subsequent heuristics in the same way. In its original form (Table 1a), the saturated cost partitioning algorithm computes an operator cost partitioning and always preserves the heuristic estimates for all states under the remaining cost function (Seipp and Helmert 2014). A generalization of saturated operator cost partitioning is saturated transition cost partitioning (Table 1b) where costs are assigned directly to transitions (Keller et al. 2016). Transition cost partitioning is more general than operator cost partitioning because each operator can induce multiple transitions. While there is no dominance relation between the operator and transition version of saturated cost partitioning, Keller et al. (2016) showed empirically that the transition version often yields more accurate heuristics than the operator version. However, our experiments show that the computational overhead caused by allowing more expressive cost assignments is often too large for the additional heuristic accuracy to lead to solving more tasks. More recently, Seipp and Helmert (2019) generalized saturated operator cost partitioning in a different direction. They allow the saturated cost partitioning algorithm to preserve the heuristic estimates of only a subset of states (Table 1c). Their empirical evaluation shows that this approach yields more accurate heuristics and achieves state-of-the-art results on the benchmark set of the International Planning Competition (IPC). To unify these two generalizations, we introduce subsetsaturated transition cost partitioning (Table 1d). It assigns costs to individual transitions and preserves the heuristic estimates of only a subset of states. We show that subsetsaturated transition cost partitioning preserves admissibility and that it usually yields stronger heuristics than the two generalizations alone.