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

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.

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

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

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

26. Simple Independence Detection Problem  Are these agents independent? From (Standley AAAI-10)

27. Simple Independence Detection Problem  Are these agents independent? From (Standley AAAI-10)

28. Better Independence Detection  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 From (Standley AAAI-10)

29. Best Independence Detection  How can we make agent 2 take this path initially? From (Standley AAAI-10)

30. Best Independence Detection  Try to avoid future conflicts  avoid the current paths of other agents. From (Standley AAAI-10)

31. 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

32. Reservation Tables Illegal move table. From (Standley AAAI-10)

33. Reservation Tables Illegal move table. From (Standley AAAI-10)

34. Reservation Tables Illegal move table. From (Standley AAAI-10)

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 Conflict avoidance table. From (Standley AAAI-10)

37. Reservation Tables Conflict avoidance table. From (Standley AAAI-10)

38. Reservation Tables Conflict avoidance table. From (Standley AAAI-10)

39. 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.

40. Complete Approximation Algorithms  Modifications to the centralized algorithm  Expand nodes with fewest violations first  Use cost to break ties

41. 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

42. Parameterized Approximation  Maximum group size parameter x  Drop constraints to avoid creating groups larger than x.  x =1 : always drop the constraints.  x = ∞ : never drop the constraints (optimal)  The algorithm is complete for any choice of x

43. 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.

44. Simple Optimal Anytime Algorithm Problem  The simple anytime algorithm suffers the cost of unused and incomplete iterations.

45. 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.

46. Optimal Anytime Algorithm  Keep a lower bound for each group.  When merging a group, add lower bounds

47. 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, we’re done

48. 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.