# From Non-Negative to General Operator Cost Partitioning

AAAI, pp. 3335-3341, 2015.

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Abstract:

Operator cost partitioning is a well-known technique to make admissible heuristics additive by distributing the operator costs among individual heuristics. Planning tasks are usually defined with non-negative operator costs and therefore it appears natural to demand the same for the distributed costs. We argue that this requirement is not...More

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Introduction

- Heuristic search is commonly used to solve classical planning tasks. Optimal planning requires admissible heuristics, which estimate the cost to the goal without overestimation.
- The authors demonstrate that when allowing negative operator costs, heuristics based on the recently proposed operator-counting constraints (Pommerening et al 2014b) can be interpreted as a form of optimal cost partitioning
- This includes the state equation heuristic (Bonet and van den Briel 2014), which was previously thought (Bonet 2013) to fall outside the four main categories of heuristics for classical planning: abstractions, landmarks, delete-relaxations and critical paths (Helmert and Domshlak 2009).

Highlights

- Heuristic search is commonly used to solve classical planning tasks
- We show that the state equation heuristic computes a general optimal cost partitioning over atomic projection heuristics
- We implemented the state equation heuristic and the potential heuristic that optimizes the heuristic value of the initial state in the Fast Downward planning system (Helmert 2006)
- We showed that the traditional restriction to non-negative cost partitioning is not necessary and that heuristics can benefit from permitting operators with negative cost
- We demonstrated that heuristics based on operatorcounting constraints compute an optimal general cost partitioning
- We introduced potential heuristics as a fast alternative to optimal cost partitioning

Results

- The authors implemented the state equation heuristic and the potential heuristic that optimizes the heuristic value of the initial state in the Fast Downward planning system (Helmert 2006).
- Previous experiments (Pommerening et al 2014b) showed that the state equation heuristic outperforms the optimal non-negative cost partitioning of projections to goal variables.
- To explain this difference the authors compare these two heuristics to the optimal cost partitioning over all projections to single variables using non-negative and general cost partitioning.
- The heuristics hOGCoaPl1+ and hOACll1P+ compute the same value because non-goal variables cannot contribute to the heuristic with non-negative cost partitioning

Conclusion

- The authors showed that the traditional restriction to non-negative cost partitioning is not necessary and that heuristics can benefit from permitting operators with negative cost.
- The authors demonstrated that heuristics based on operatorcounting constraints compute an optimal general cost partitioning.
- This allows for a much more compact representation of cost-partitioning LPs, and the authors saw that the state equation heuristic can be understood as such a compact form of expressing an optimal cost partitioning over projections.
- The authors introduced potential heuristics as a fast alternative to optimal cost partitioning.
- The authors think that the introduction of potential heuristics opens the door for many interesting research avenues and intend to pursue this topic further in the future

Summary

## Introduction:

Heuristic search is commonly used to solve classical planning tasks. Optimal planning requires admissible heuristics, which estimate the cost to the goal without overestimation.- The authors demonstrate that when allowing negative operator costs, heuristics based on the recently proposed operator-counting constraints (Pommerening et al 2014b) can be interpreted as a form of optimal cost partitioning
- This includes the state equation heuristic (Bonet and van den Briel 2014), which was previously thought (Bonet 2013) to fall outside the four main categories of heuristics for classical planning: abstractions, landmarks, delete-relaxations and critical paths (Helmert and Domshlak 2009).
## Results:

The authors implemented the state equation heuristic and the potential heuristic that optimizes the heuristic value of the initial state in the Fast Downward planning system (Helmert 2006).- Previous experiments (Pommerening et al 2014b) showed that the state equation heuristic outperforms the optimal non-negative cost partitioning of projections to goal variables.
- To explain this difference the authors compare these two heuristics to the optimal cost partitioning over all projections to single variables using non-negative and general cost partitioning.
- The heuristics hOGCoaPl1+ and hOACll1P+ compute the same value because non-goal variables cannot contribute to the heuristic with non-negative cost partitioning
## Conclusion:

The authors showed that the traditional restriction to non-negative cost partitioning is not necessary and that heuristics can benefit from permitting operators with negative cost.- The authors demonstrated that heuristics based on operatorcounting constraints compute an optimal general cost partitioning.
- This allows for a much more compact representation of cost-partitioning LPs, and the authors saw that the state equation heuristic can be understood as such a compact form of expressing an optimal cost partitioning over projections.
- The authors introduced potential heuristics as a fast alternative to optimal cost partitioning.
- The authors think that the introduction of potential heuristics opens the door for many interesting research avenues and intend to pursue this topic further in the future

- Table1: Coverage for different variants of optimal cost partitioning

Funding

- This work was supported by the Swiss National Science Foundation (SNSF) as part of the project “Abstraction Heuristics for Planning and Combinatorial Search” (AHPACS)

Reference

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