Saturated cost partitioning (SCP) is one of the strongest methods for admissibly combining heuristics for optimal classical planning. The quality of an SCP heuristic depends heavily on the order in which its component heuristics are considered. For high accuracy, it is essential to maximize over multiple SCP heuristics computed using different component orders. However, for n component heuristics, even enumerating all n! orders is usually infeasible. Consequently, previous work resorted to using greedy algorithms and local optimization. In contrast, we present the first practical method for computing the perfect SCP heuristic that is equivalent to considering all component orders. We show that a set of SCP heuristics forms an additive disjunctive heuristic, which allows us to concisely represent component orders as a directed acyclic graph. Furthermore, once certain components have been considered, the order of the remaining components often becomes irrelevant. By exploiting this characteristic, we can reduce the size of the heuristic representation by several orders of magnitude in practice. Finally, our work makes it possible to compare the quality of existing SCP methods with that of the perfect SCP heuristic, revealing that existing approximations are nearly optimal for standard benchmarks.

Saturated post-hoc optimization is a powerful method for computing admissible heuristics for optimal classical planning. The approach solves a linear program (LP) for each state encountered during the search, which is computationally demanding. In this paper, we theoretically and empirically analyze to which extent we can reuse an LP solution of one state for another. We introduce a novel sensitivity analysis that can exactly characterize the set of states for which a unique LP solution is optimal. Furthermore, we identify two properties of the underlying LPs that affect reusability. Finally, we introduce an algorithm that optimizes LP solutions to generalize well to other states. Our new algorithms significantly reduce the number of necessary LP computations.

Matrix multiplication is a fundamental operation of linear algebra, with applications ranging from quantum physics to artificial intelligence. Given its importance, enormous resources have been invested in the search for faster matrix multiplication algorithms. Recently, this search has been cast as a single-player game. By learning how to play this game efficiently, the newly-introduced AlphaTensor reinforcement learning agent is able to discover many new faster algorithms. In this paper, we show that finding matrix multiplication algorithms can also be cast as a classical planning problem. Based on this observation, we introduce a challenging benchmark suite for classical planning and evaluate state-of-the-art planning techniques on it. We analyze the strengths and limitations of different planning approaches in this domain and show that we can use classical planning to find lower bounds and concrete algorithms for matrix multiplication.

Cost partitioning is the foundation of today’s strongest heuristics for optimal classical planning. However, computing a cost partitioning for each evaluated state is prohibitively expensive in practice. Thus, existing approaches make an approximation and compute a cost partitioning only for a set of sampled states, and then reuse the resulting heuristics for all other states evaluated during the search. In this paper, we present exact methods for cost partitioning heuristics based on linear programming that fully preserve heuristic accuracy while minimizing computational cost. Specifically, we focus on saturated post-hoc optimization and establish several sufficient conditions for when reusing a cost partitioning computed for one state preserves the estimates for other states, mainly based on a sensitivity analysis of the underlying linear program. Our experiments demonstrate that our theoretical results transfer into practice, and that our exact cost partitioning algorithms are competitive with the strongest approximations currently available, while usually requiring fewer linear program evaluations.