Pruning techniques based on partial order reduction and symmetry elimination have recently found increasing attention for optimal planning. Although these techniques appear to be rather different, they base their pruning decisions on similar ideas from a high level perspective. In this paper, we propose safe integrations of partial order reduction and symmetry elimination for cost-optimal classical planning. We show that previously proposed symmetry-based search algorithms can safely be applied with strong stubborn sets. In addition, we derive the notion of symmetrical strong stubborn sets as a more tightly integrated concept. Our experiments show the potential of our approaches.

Merge-and-shrink heuristics crucially rely on effective reduction techniques, such as bisimulation-based shrinking, to avoid the combinatorial explosion of abstractions. We propose the concept of factored symmetries for merge-and-shrink abstractions based on the established concept of symmetry reduction for state-space search. We investigate under which conditions factored symmetry reduction yields perfect heuristics and discuss the relationship to bisimulation. We also devise practical merging strategies based on this concept and experimentally validate their utility.

Heuristic search is a state-of-the-art approach to classical planning. Several heuristic families were developed over the years to automatically estimate goal distance information from problem descriptions. Orthogonally to the development of better heuristics, recent years have seen an increasing interest in symmetry-based state space pruning techniques that aim at reducing the search effort. However, little work has dealt with how the heuristics behave under symmetries. We investigate the symmetry properties of existing heuristics and reveal that many of them are invariant under symmetries.

Hartigan's method for k-means clustering holds several potential advantages compared to the classical and prevalent optimization heuristic known as Lloyd's algorithm. E.g., it was recently shown that the set of local minima of Hartigan's algorithm is a subset of those of Lloyd's method. We develop a closed-form expression that allows to establish Hartigan's method for k-means clustering with any Bregman divergence, and further strengthen the case of preferring Hartigan's algorithm over Lloyd's algorithm. Specifically, we characterize a range of problems with various noise levels of the inputs, for which any random partition represents a local minimum for Lloyd's algorithm, while Hartigan's algorithm easily converges to the correct solution. Extensive experiments on synthetic and real-world data further support our theoretical analysis.