Finding Optimal Solutions to Cooperative Pathfinding Problems Trevor Standley Computer Science Department University of California, Los Angeles 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 Motivation My Formulation Gridworld pathfinding Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal Related Work Centralized Approaches Strengths: Typically complete, can be optimal Weaknesses: Takes forever! Decoupled Approaches Strengths: Fast Weaknesses: Incomplete and suboptimal 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 My algorithm Optimal Complete Two main contributions Operator decomposition Independence detection 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 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 Operator Decomposition Example Operator Decomposition The Savings of Operator Decomposition Consequences of Operator Decomposition Branching factor becomes polynomial However, state space still exponential Simple Independence Detection 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 Simple Independence Detection Simple Independence Detection Problem Are these agents independent? Simple Independence Detection Problem Are these agents independent? 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 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 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 Results randomly generated problems with 2-60 agents 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 Acknowledgments My advisor, Rich Korf. Dawn Chen for editing, advice, and artwork