Disjoint Splitting for Multi-Agent Path Finding with Conflict-Based - - PowerPoint PPT Presentation

Disjoint Splitting for Multi-Agent Path Finding with Conflict-Based Search

Jiaoyang Li, Daniel Harabor, Peter J. Stuckey, Ariel Felner, Hang Ma, and Sven Koenig Berkeley, CA 07/15/2019

Multi-Agent Path Finding (MAPF)

• Goal:
• Find collision-free paths.
• Minimize the sum of path costs.
• MAPF is NP-hard to solve optimally.

1 2 1 2

Conflict-Based Search (CBS)

Plan paths independently for each agent.

A B C

1 2 1 2

Check for conflicts. Resolve a conflict by adding new constraints. Both agents are at C at time 3.

Conflict-Based Search (CBS)

OR

Case 1: Agent 1 cannot be at C at time 3. Case 2: Agent 2 cannot be at C at time 3. Plan paths independently for each agent. Check for conflicts. Resolve a conflict by adding new constraints. Replan paths that satisfy the constraints. Repeat until finding collision-free paths.

1 2 1

A B C

1 2 2

A B C

Conflict-Based Search (CBS)

• The high-level of CBS searches in a binary tree using

a best-first manner.

• The low-level of CBS replans paths for single agents.

The splitting of CBS

• To resolve a conflict between agents 𝑗 and 𝑘 at vertex 𝑤 at time 𝑢:
• The searching spaces (i.e., sets of paths that satisfy the constraints) of

the two child nodes are not disjoint!

• The following pair of paths satisfies both constraints.

𝐷 ∪ 𝑗, 𝑤, 𝑢 𝐷 ∪ 𝑘, 𝑤, 𝑢 ⟨𝑗, 𝑤, 𝑢⟩ : Agent 𝑗 cannot be at vertex 𝑤 at time 𝑢. 𝐷 Time … 𝑢 … Path for 𝑗

𝑡𝑗 … 𝑤′ … 𝑕𝑗

Path for 𝑘

𝑡

𝑘

… 𝑤′′ … … … 𝑕𝑘

The splitting of CBS

• Non-disjoint splitting:
• Disjoint splitting:

𝐷 ∪ 𝑗, 𝑤, 𝑢 𝐷 ∪ 𝑘, 𝑤, 𝑢 𝐷 𝐷 ∪ 𝑗, 𝑤, 𝑢 𝐷 ∪ ⟨𝑗, 𝑤, 𝑢⟩ 𝐷

• Negative constraint ⟨𝑗, 𝑤, 𝑢⟩ :
• Agent 𝑗 cannot be at 𝑤

at time 𝑢.

• Positive constraint 𝒋, 𝒘, 𝒖 :
• Agent 𝑗 must be at 𝑤 at time 𝑢.
• Any other agents (including

agent 𝑘) cannot be at 𝑤 at time 𝑢.

Example

A B C

1 2 1 2

⟨1, 𝐷, 3⟩ ⟨2, 𝐷, 3⟩ ⟨1, 𝐶, 2⟩ ⟨2, 𝐶, 2⟩ ⟨1, 𝐵, 1⟩ ⟨2, 𝐵, 1⟩

8 9 8 9 9 9 8

• Non-disjoint splitting:

Example

A B C

1 2 1 2

⟨1, 𝐷, 3⟩ ⟨2, 𝐷, 3⟩ ⟨1, 𝐶, 2⟩ ⟨2, 𝐶, 2⟩ ⟨1, 𝐵, 1⟩ ⟨2, 𝐵, 1⟩ ⟨1, 𝐷, 3⟩

8 9 8 9 9 9 8

⟨1, 𝐷, 3⟩ ⟨1, 𝐶, 2⟩ ⟨1, 𝐵, 1⟩

8 9 8 9 8

⟨1, 𝐵, 1⟩ ⟨1, 𝐶, 2⟩

• Non-disjoint splitting:
• Disjoint splitting:

Example

A B C

1 2 1 2

⟨1, 𝐷, 3⟩ ⟨2, 𝐷, 3⟩ ⟨1, 𝐶, 2⟩ ⟨2, 𝐶, 2⟩ ⟨1, 𝐵, 1⟩ ⟨2, 𝐵, 1⟩ ⟨1, 𝐷, 3⟩

8 9 8 9 9 9 8

⟨1, 𝐷, 3⟩ ⟨1, 𝐶, 2⟩ ⟨1, 𝐵, 1⟩

8 9 8 9 8

⟨1, 𝐵, 1⟩ ⟨1, 𝐶, 2⟩

• Non-disjoint splitting:
• Disjoint splitting:

Low-level search of CBS

• Non-disjoint splitting
• Replan the entire path.
• Disjoint splitting
• Replan the path segment between two positive constraints.

Experiments

3 22 142 1,015 7,447 62,429 573,004 3 11 38 110 339 935 2,352 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 2X2 3X3 4X4 5X5 6X6 7X7 8X8 Nodes Rectangle size Non-disjoint Disjoint

1 2 1 2

Expanded nodes

2x2 rectangle conflict

Experiments

0.00 0.20 0.40 0.60 0.80 1.00 10 20 30 40 50 60 Success Rate Agents 0.00 0.20 0.40 0.60 0.80 1.00 20 30 40 50 60 Success Rate Agents 0.00 0.20 0.40 0.60 0.80 1.00 10 12 14 16 18 20 Success Rate Agents 0.00 0.20 0.40 0.60 0.80 1.00 20 40 60 80 100 120 140 Success Rate Agents

Success rate (%solved instances within 5 minutes)

Takeaways

• The splitting of standard CBS is not disjoint.
• Disjoint splitting:
• Empirically, disjoint splitting is at least as good as CBS splitting and

significantly speeds up CBS in many cases. 𝐷 ∪ 𝑗, 𝑤, 𝑢 𝐷 ∪ ⟨𝑗, 𝑤, 𝑢⟩ 𝐷

Positive constraint:

• Agent 𝑗 must be at 𝑤 at

time 𝑢.

• Any other agents cannot

be at 𝑤 at time 𝑢.