NBS applied to Planning

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NBS NBS Experiment fMM Experiment Conclusion

NBS applied to Planning

Marvin Buff

University of Basel

11. February, 2019

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Motivation

Foundation

Classical Planning: Devise a new efficient and competitive algorithm.

Inspiration

The introduction of a new and promising bidirectional search algorithm: Near-Optimal Bidirectional Search Algorithm (NBS).

Realization

Implementing NBS as planner.

Evaluating its performance on the IPC benchmark.

Analysing the results with fMM.

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Motivation

Foundation

Inspiration

Realization

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Motivation

Foundation

Inspiration

Realization

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Outline

1 NBS

The Blocks Domain Sufficient Conditions NBS in Detail

2 NBS Experiment Running NBS

Results and Evaluation

3 fMM Experiment MM

fMM Properties fMM Experiment Case Study

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Outline

NBS

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The Blocks Domain

Table Table

A B C A

B C

Actions Pick Up Put Down

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The Blocks Domain

Transition System

A B C

A B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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Sufficient Conditions

Surely Expanded States: f (s) < C^∗= 2

A B C

A B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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Sufficient Conditions

Bidirectional Search: ?

A B C

A B

C A C

B B A

C B C

A A C

B A B

C

A B

C A C

B B A

C B C

A A C

B A B

C A

B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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Sufficient Conditions

A B C

A B

C A C

B B A

C B C

A A C

B A B

C

A B

C A C

B B A

C B C

A A C

B A B

C

A B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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Sufficient Conditions

A B C

A B

C A C

B B A

C B C

A A C

B A B

C A

B

C A C

B B A

C B C

A A C

B A B

C

A B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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Sufficient Conditions

Unidirectional Search: Sufficient Condition g (s) + h(s) < C^∗

s₀ s s_G

C^∗

Bidirectional Search: Must-Expand Pairs

max {f_F(u), f_B(v ), g (u) + g (v )} < C^∗

s₀ u v s_G

C^∗

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Sufficient Conditions

Unidirectional Search: Sufficient Condition g (s) + h(s) < C^∗

s₀ s s_G

C^∗ Bidirectional Search: Must-Expand Pairs

max {f_F(u), f_B(v ), g (u) + g (v )} < C^∗

s₀ u v s_G

C^∗

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Sufficient Conditions

Must-Expand Graph: G_MX

A B C

?

A B C

?

? A B C

6x 6x

g_F = 0

g_F = 1

g_F = 2

g_B = 0

g_B = 1

g_B = 2

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Sufficient Conditions

Minimal Vertex Cover of G_MX (VC)

A B C

?

A B C

?

? A B C

6x 6x

gF = 0

g_F = 1

gF = 2

gB = 0

g_B = 1

gB = 2

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Sufficient Conditions

Bidirectional Search: Optimal Expansion

A B C

A B

C A C

B B A

C B C

A A C

B A B

C

A B C

C B A

B A C

C A B

A C B

B C A

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NBS

Algorithm 1 NBS

1: Put s0 in Open_F and s_G in Open_B

2: while Open_F and Open_B are not empty do

3: Among u ∈ OpenF and v ∈ OpenB 4: Select the pair (u, v ) with lowest lb(u, v )

5: if lb(u, v ) ≥ cost(U) then

6: return U

7: Expand both u and v

8: if new path from s₀ to s_G is found then

9: Update U if new path is better than previous

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NBS

Key Properties

Admissible DXBB Algorithm Optimal Worst-Case

Near-Optimal General Case

Key Difference to Search

Backward search is not a mirrored forward search.

Actions are not trivially reversible.

There are Secondary Initial States.

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NBS

Key Properties

Admissible DXBB Algorithm Optimal Worst-Case

Near-Optimal General Case

Key Difference to Search

Backward search is not a mirrored forward search.

Actions are not trivially reversible.

There are Secondary Initial States.

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Outline

NBS Experiment

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NBS Experiment

Goal

Evaluate performance of NBS compared to A^∗. Parameters

Latest IPC Benchmark

Implementation in Fast Downward

Time limit: 30 minutes. Memory Limit: 3.5 GB.

Heuristics

Blind Heuristic Max Heuristic

Critical Path Heuristic (m = 2) Landmark-Cut Heuristic

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NBS Experiment

Goal

Evaluate performance of NBS compared to A^∗. Parameters

Latest IPC Benchmark

Implementation in Fast Downward

Time limit: 30 minutes. Memory Limit: 3.5 GB.

Heuristics

Blind Heuristic Max Heuristic

Critical Path Heuristic (m = 2) Landmark-Cut Heuristic

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Results and Evaluation

NBS > A^∗

Domain Solved Problems (#) Algo h¹ h^max h^m h^lmcut

Blocks A^∗ 18 21 10 28

NBS 27 27 13 31

Similar Domains (∼ 10) Driverlog

Floortile-opt11-strips ..

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Results and Evaluation

NBS > A^∗

Blocks A^∗ 18 21 10 28

NBS 27 27 13 31

Similar Domains (∼ 10) Driverlog

Floortile-opt11-strips ..

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Results and Evaluation

NBS < A^∗

Storage A^∗ 14 15 7 15

NBS 5 5 2 4

Similar Domains (∼ 20) Depot

Elevators-opt11-strips ..

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Results and Evaluation

NBS < A^∗

Storage A^∗ 14 15 7 15

NBS 5 5 2 4

Similar Domains (∼ 20) Depot

Elevators-opt11-strips ..

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Outline

fMM Experiment

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Meet-In-The-Middle

State Space

Dijkstra

MM MM

s₀ s_G

C ∗

2 C ∗

2 C^∗

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Fractional Meet-In-The-Middle

p = 0.5

Dijkstra

fMM fMM

s0 sG

p · C^∗ (1 − p) · C^∗

C^∗

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Fractional Meet-In-The-Middle

p = 0.5

s₀ s_G

p · C^∗ (1 − p) · C^∗

p = 0.25

s₀ s_G

p · C^∗ (1 − p) · C^∗

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Fractional Meet-In-The-Middle

Key Properties

Expands |VC| states given input p^∗. Reflects the structure of the state space.

Experiment Goal

Inquire the relation between the state space structure and the performance of NBS.

Experiment Parameters

Same constraints as the NBS experiment.

Equidistant p-value from 0 to 1 with step size 0.1.

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Fractional Meet-In-The-Middle

Key Properties

Experiment Goal

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Fractional Meet-In-The-Middle

Key Properties

Experiment Goal

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Results and Evaluation

Backward Forward

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Results and Evaluation

Backward Forward

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Results and Evaluation

Overview

p^∗ is informative of the overall structure of the state space.

The state space influences the performance of NBS.

The structure of the state space correlates with the domain.

How can we use that information?

Case Study: Blocks Domain

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Results and Evaluation

Overview

p^∗ is informative of the overall structure of the state space.

The state space influences the performance of NBS.

The structure of the state space correlates with the domain.

How can we use that information?

Case Study: Blocks Domain

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Case Study: Blocks Instance 6-1

Table Table

B C D E F

A

D F E A B C

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Case Study: Blocks Instance 6-1

Backward Forward

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Case Study: Blocks Instance 6-0

Table Table

B E F

C A D

D F E A B C

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Case Study: Blocks Instance 6-0

Backward Forward

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Case Study: Blocks Instance 6-2

Table Table

C E F B D A

D C B A F E

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Case Study: Blocks Instance 6-2

Backward Forward

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Conclusion

Take Home Message

Using NBS in planning can compete with A^∗.

The efficiency of NBS depends on the structure of the transition system.

The predisposition towards certain search approaches correlates with the domain.

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Conclusion

Take Home Message

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Conclusion

Take Home Message

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Questions?

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Appendix

For Further Reading I

J. Chen, R. C. Holte, S. Zilles, N. Sturtevant

Front-to-end bidirectional heuristic search with near-optimal node expansions.

IJCAI 2017, pages: 489-495, AAAI Press.

J. Eckerle, J. Chen, R. C. Holte, S. Zilles

Sufficient conditions for node expansion in bidirectional heuristic search.

ICAPS 2017, pages: 79-87, AAAI Press.

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Gripper Instance 06

Backward Forward

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Appendix

Logistics00 Instance 4-0

Backward Forward

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Openstacks-Strips Instance 01

Backward Forward

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Appendix

Termes Instance 03

Backward Forward