Finding Optimal Solutions to Cooperative Pathfinding Problems

1
Finding Optimal Solutions to Cooperative
Pathfinding Problems

• Trevor Standley and Rich Korf
• Computer Science Department
• University of California, Los Angeles

2
Introduction

• Pathfinding Problems
• A single agent must find a path from a start
  state to a goal state
• Cooperative Pathfinding Problems
• Multiple agents interact
• Want to minimize the total cost

3
Motivation
4
Motivation
5
My Formulation

• Gridworld pathfinding

6
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

7
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

8
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

9
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

10
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

11
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

12
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

13
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

14
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

15
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

16
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

17
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

18
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

19
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

20
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

21
Related Work

• Centralized Approaches
• Strengths Typically complete, can be optimal
• Weaknesses Takes forever!
• Decoupled Approaches
• Strengths Fast
• Weaknesses Incomplete and suboptimal

22
Our Prior Work (Standley AAAI-10)

• Independence Detection
• Empowers centralized algorithms.
• Combines the strength of centralized and
  decentralized approaches.
• Maintains optimality and completeness.

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Simple Independence Detection
From (Standley AAAI-10)
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Simple Independence Detection
From (Standley AAAI-10)

1. Put each agent into its own group.
2. Plan paths for each group independently
3. Check for conflicts in new paths
4. Combine groups with conflicting paths
5. Repeat 2-4 until no conflicts

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Simple Independence Detection
From (Standley AAAI-10)
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Simple Independence Detection Problem
From (Standley AAAI-10)

• Are these agents independent?

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Simple Independence Detection Problem
From (Standley AAAI-10)

• Are these agents independent?

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Better Independence Detection
From (Standley AAAI-10)

• When a conflict is detected between two groups,
  try to find an alternative path for one of the
  groups
• If that fails try to find an alternate path for
  the other group
• Only as a last resort do we combine the groups

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Best Independence Detection
From (Standley AAAI-10)

• How can we make agent 2 take this path initially?

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Best Independence Detection
From (Standley AAAI-10)

• Try to avoid future conflicts
• avoid the current paths of other agents.

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Reservation Tables

• Illegal move table
• Contains all the ways alternative paths could
  result in a conflict with the currently
  conflicting group.
• Consider such moves illegal.
• Conflict avoidance table
• Contains all the ways alternative paths could
  result in a conflict with any other group
• Keep track of conflict avoidance table violations
  and

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Reservation Tables
From (Standley AAAI-10)

• Illegal move table.

33
Reservation Tables
From (Standley AAAI-10)

• Illegal move table.

34
Reservation Tables
From (Standley AAAI-10)

• Illegal move table.

35
Reservation Tables

• Illegal move table
• Contains all the ways alternative paths could
  result in a conflict with the currently
  conflicting group.
• Consider such moves illegal.
• Conflict avoidance table
• Contains all the ways alternative paths could
  result in a conflict with any other group
• Keep track of conflict avoidance table violations

36
Reservation Tables
From (Standley AAAI-10)

• Conflict avoidance table.

37
Reservation Tables
From (Standley AAAI-10)

• Conflict avoidance table.

38
Reservation Tables
From (Standley AAAI-10)

• Conflict avoidance table.

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Complete Approximation Algorithms

• Our previous work maintained optimality by
• Only accepting alternate paths if they have the
  same cost as original paths.
• Coupling independence detection with an optimal
  centralized algorithm.
• We recognize in our current work that we can drop
  these two constraints.

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Complete Approximation Algorithms

• Modifications to the centralized algorithm
• Expand nodes with fewest violations first
• Use cost to break ties

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When to drop these constraints

• Always
• Leads to a fast and complete algorithm
• When doing so avoids the creation of groups
  containing more than x agents
• Leads to a slower but still fast algorithm
• Produces higher quality paths

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Parameterized Approximation

• Maximum group size parameter x
• Drop constraints to avoid creating groups larger
  than x.
• x 1 always drop the constraints.
• x 8 never drop the constraints (optimal)
• The algorithm is complete for any choice of x

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Simple Optimal Anytime Algorithm

• Run the parameterized approximation with x 1.
• Then run the parameterized approximation with x
  2.
• When we run out of time, we return the best
  solution found by any run.

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Simple Optimal Anytime Algorithm Problem

• The simple anytime algorithm suffers the cost of
  unused and incomplete iterations.

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Optimal Anytime Algorithm Problem

• Keep paths and groupings from previous iterations
  when possible.
• Keep track of groups that might not have optimal
  paths.
• Fix these paths one at a time starting with the
  easiest.

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Optimal Anytime Algorithm

• Keep a lower bound for each group.
• When merging a group, add lower bounds

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Optimal Anytime Algorithm

• Update best path many times within an iteration.
• Whenever the solution is conflict free we update
  the best solution found.
• When lower bound equals cost, were done

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Results

• Our coarsest approximation is complete, has
  competitive running time, and produces superior
  solutions.
• As an optimal algorithm, our anytime algorithm is
  competitive with our previous state-of-the-art.
• If our anytime algorithm is terminated early, it
  often returns an optimal path.