1. Finding Optimal Solutions to Cooperative Pathfinding Problems Trevor Standley Computer Science Department University of California, Los Angeles http://cs.ucla.edu/~tstand/

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

6. My Formulation  Gridworld pathfinding

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. The Standard Algorithm  The standard algorithm is A*  Centralized algorithm  There is a standard heuristic  State representation – A position for each agent  State space – Exponential in the number of agents  An operator – Complete assignment of moves to agents -One of {N; NE; E; SE; S; SW; W; NW; and wait} for each agent -Exponential in the number of agents  Obviously this algorithm is not taken seriously

23. My algorithm  Optimal  Complete  Two main contributions  Operator decomposition  Independence detection

24. Operator Decomposition  Intuition  Also a centralized algorithm  Still use A*  Change how operators are defined: only one agent moves at a time  Simple idea, tricky to get details right

25. Operator Decomposition  Each operator assigns a move to a single agent  Assignments are made in a fixed order  Move assignments stored as part of the state representation

26. Operator Decomposition  Example

27. Operator Decomposition

28. The Savings of Operator Decomposition

29. Consequences of Operator Decomposition  Branching factor becomes polynomial  However, state space still exponential

30. Simple Independence Detection

31. 1.Create a group for each agent 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

32. Simple Independence Detection

33. Simple Independence Detection Problem  Are these agents independent?

34. Simple Independence Detection Problem  Are these agents independent?

35. Better Independence Detection  When a conflict is detected between two groups, try to find an alternate path for one of the groups  If that fails try to find an alternate path for the other group  Only combine groups if no alternate path could be found

36. Independence Detection  Which alternate paths are the best?  Only search for optimal paths  Paths can be found using operator decomposition  Find paths that will lead to fewest number of future conflicts  Operator decomposition can be modified to find optimal paths with few future conflicts

37. My Algorithm  Uses decoupled planning where possible  Only uses centralized planning for non-independent subproblems  Calls operator decomposition as a subroutine to do the centralized planning

38. Results  10000 randomly generated problems with 2-60 agents

39. Conclusions  Researchers have developed centralized and decoupled approaches for solving cooperative pathfinding problems  Operator decomposition is an improved centralized approach  Independence detection is a hybrid approach  Only uses centralized planning when necessary

40. Acknowledgments  My advisor, Rich Korf.  Dawn Chen for editing, advice, and artwork