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Title: Pattern Selection for Optimal Classical Planning with Saturated Cost Partitioning

Paper content:
Pattern databases are the foundation of some of the strongest admissible heuristics for optimal classical planning.
Experiments showed that the most informative way of combining information from multiple pattern databases is to use saturated cost partitioning.
Previous work selected patterns and computed saturated cost partitionings over the resulting pattern database heuristics in two separate steps.
We introduce a new method that uses saturated cost partitioning to select patterns and show that it outperforms all existing pattern selection algorithms.
A * search BID10 with an admissible heuristic BID23 ) is one of the most successful methods for solving classical planning tasks optimally.
An important building block of some of the strongest admissible heuristics are pattern database (PDB) heuristics.
A PDB heuristic precomputes all goal distances in a simplified state space obtained by projecting the task to a subset of state variables, the pattern, and uses these distances as lower bounds on the true goal distances.
PDB heuristics were originally introduced for solving the 15-puzzle BID2 and have later been generalized to many other combinatorial search tasks (e.g., BID21 BID7 and to the setting of domainindependent planning BID3 .Using a single PDB heuristic of reasonable size is usually not enough to cover sufficiently many aspects of challenging planning tasks. It is therefore often beneficial to compute multiple PDB heuristics and to combine their estimates admissibly BID15 . The simplest approach for this is to choose the PDB with the highest estimate in each state. Instead of this maximization scheme, we would like to sum estimates, but this renders the resulting heuristic inadmissible in general. However, if two PDBs are affected by disjoint sets of operators, they are independent and we can admissibly add their estimates BID19 BID7 . BID11 later generalized this idea by introducing the canonical heuristic for PDBs, which computes all maximal subsets of pairwise independent PDBs and then uses the maximum over the sums of independent PDBs as the heuristic value.Cost partitioning BID17 BID40 ) is a generalization of the independence-based methods above.
It makes the sum of heuristic estimates admissible by distributing the costs of each operator among the heuristics.
The literature contains many different cost partitioning algorithms such as zero-one cost partitioning BID4 BID11 ), uniform cost partitioning BID17 , optimal cost partitioning BID17 BID16 BID18 BID25 , posthoc optimization BID26 and delta cost partitioning BID6 .In
previous work BID34 , we showed experimentally for the benchmark tasks from previous International Planning Competitions (IPC) that saturated cost partitioning (SCP) BID30 BID37 is the cost partitioning algorithm of choice for PDB heuristics. Saturated
cost partitioning considers an ordered sequence of heuristics. Iteratively
, it gives each heuristic the minimum amount of costs that the heuristic needs to justify all its estimates and then uses the remaining costs for subsequent heuristics until all heuristics have been served this way.Before we can compute a saturated cost partitioning over pattern database heuristics, we need to select a collection of patterns. The first
domain-independent automated pattern selection algorithm is due to BID3 . It partitions
the state variables into patterns via best-fit bin packing. BID5 later used
a genetic algorithm to search for a pattern collection that maximizes the average heuristic value of a zero-one cost partitioning over the PDB heuristics. BID11 proposed
an algorithm that performs a hill-climbing search in the space of pattern collections (HC). HC evaluates a
collection C by estimating the search effort of the canonical heuristic over C based on a model of IDA * runtime BID20 . BID8 presented
the Complementary PDBs Creation (CPC) method, that combines bin packing and genetic algorithms to create a pattern collection minimizing the estimated search effort of an A * search BID22 . BID28 repeatedly
compute patterns using counterexample-guided abstraction refinement (CEGAR): starting from a random goal variable, their CEGAR algorithm iteratively finds solutions in the corresponding projection and executes them in the original state space. Whenever a solution
cannot be executed due to a violated precondition, it adds the missing precondition variable to the pattern.Finally, BID26 systematically generate all interesting patterns up to a given size X (SYS-X). Experiments showed
that cost-partitioned heuristics over SYS-2 and SYS-3 yield accurate estimates BID26 BID34 , but using all interesting patterns of larger sizes is usually infeasible.We introduce SYS-SCP, a new pattern selection algorithm based on saturated cost partitioning that potentially considers all interesting patterns, but only selects useful ones. SYS-SCP builds multiple
pattern sequences that together form the resulting pattern collection. For each sequence σ, it
considers the interesting patterns in increasing order by size and adds a pattern P to σ if P is not part of an earlier sequence and the saturated cost partitioning heuristic over σ plus P is more informative than the one over σ alone.
We introduced a new pattern selection algorithm based on saturated cost partitioning and showed that it outperforms Table 6 : Number of tasks solved by different planners.

assistant: text: Using saturated cost partitioning to select patterns is preferable to all existing pattern selection algorithms.