Optimal Target Assignment and Path Finding for Teams of Agents

Transcription

1 Optimal Target Assignment and Path Finding for Teams of Agents Hang Ma Sven Koenig University of Southern California May 12, 2016 AAMAS, Singapore

2 Multi-Agent Path Finding (MAPF) Find collision-free paths for all agents from their start vertices to their targets.

3 Time Step 0

4 Time Step 1

5 Time Step 2

6 Time Step 3

7 Target Assignment and Path Finding (TAPF) TAPF A mix of non-anonymous MAPF and anonymous MAPF. Non-Anonymous MAPF NP-Hard Anonymous MAPF P + = TAPF

8 An Example of TAPF Team0: Agents that move from the stations to the storage locations. Team1: Agents that move from the storage locations to Station 1. Team2: Agents that move from the storage locations to Station 2. Team3: Agents that move from the storage locations to Station Figure: Kiva (Amazon Robotics) Automated Warehouse System 1. Figure 3: A small region of a Kiva layout. The green cells represent pod storage locations, the orange ovals the robots (with pods not pictured), and the purple and pink regions the queues around the inventory stations. 1 P. R. Wurman, R. D Andrea, and M. Mountz. Coordinating Hundreds of Cooperative, Autonomous Vehicles in Warehouses. In: AI Magazine 29.1 (2008), pp used to move the inventory pods with the correct bins from their storage locations to the inventory stations where a pick worker removes the desired products from the desired bin.

9 Task of TAPF Find the target assignments and collision-free paths that minimize the makespan. The makespan = the earliest time step when all agents have reached their targets.

10 How to Solve TAPF? Ideas from: Conflict-Based Search (CBS) 2 for solving non-anonymous MAPF (NP-hard). Max-flow algorithm 3 for solving anonymous MAPF (P). 2 G. Sharon et al. Conflict-based search for optimal multi-agent pathfinding. In: Artificial Intelligence 219 (2015), pp J. Yu and S. M. LaValle. Multi-agent Path Planning and Network Flow. In: Algorithmic Foundations of Robotics X, Springer Tracts in Advanced Robotics. Vol , pp

11 Conflict-Based Min-Cost Flow (CBM) for TAPF Our algorithm Conflict-Based Min-Cost Flow (CBM) = Conflict-Based Search (CBS) + (min-cost) max flow

12 CBS for Non-Anonymous MAPF CBS: 1. Find paths for each single agent separately. 2. Look for collisions in paths. 3. If there is a collision between a 1 and a 2 : Option 1 or Option 2 to avoid collision. Collision: Agent a 1, Agent a 2, Location x, Time t Constraint: Agent, Location, Time Option 1: a 1 cannot stay in x at time step t. Option 2: a 2 cannot stay in x at time step t.

13 Conflict-Based Min-Cost Flow (CBM) for TAPF Our algorithm: Conflict-Based Min-Cost Flow (CBM) considers each team to be a meta-agent. A best-search on a search tree, nodes stored in a priority queue. The key of each tree node is the makespan of the paths stored in the node. For every tree node: 1. Find paths for a single team separately. 2. Look for collisions in paths. 3. If there is a collision between team 1 and team 2 : Option 1 or Option 2 to avoid collision.

14 Conflict-Based Min-Cost Flow (CBM) for TAPF Collision: Team team 1, Team team 2, Location x, Time t Constraint: Team, Location, Time Option 1: agents in team 1 cannot stay in x at time step t. Option 2: agents in team 2 cannot stay in x at time step t.

15 Finding Paths for Single Teams Find paths for each single team team i separately = 1. Assign agents in team i to targets given to team i AND 2. Find paths for team i that have no collisions among agents in team i, according to the target assignment. Use a polynomial-time min-cost max-flow algorithm on a time-expanded network.

16 An Example c c a b d f a b d f e e

17 Finding Paths for Single Teams Separately a b d f s 2 g g 2 1 s 2 1 a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out c e s 1 1 g 1 1 team1 {c,d,e} team2 {a,b,d} {b,d,f} a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out team1 team2

18 Storing Paths and Key Root key = 2

19 Looking for Collisions in Paths a b d f s 2 g g 2 1 s 2 1 a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out c e s 1 1 g 1 1 team1 {c,d,e} team2 {a,b,d} {b,d,f} a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out team1 team2

20 Storing Colliding Teams Root key = 2 Colliding Teams (team1, team2)

21 Poping a Tree Node Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1)

22 Two Options Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) (team2, d,1)

23 Option 1: Find New Paths for team 1 a b d f s 2 g g 2 1 s 2 1 c e s 1 1 g 1 1 team1 {c,c,d,e} team2 {a,b,d} {b,d,f} Constraints (team1,d,1) a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out 3 in 3 out team1 team2

24 Storing Paths and Key Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) (team2, d,1) team1 key = 3

25 Looking for Collisions in Paths a b d f s 2 g g 2 1 s 2 1 c e s 1 1 g 1 1 team1 {c,c,d,e} team2 {a,b,d} {b,d,f} Constraints (team1,d,1) a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out 3 in 3 out team1 team2

26 Storing Colliding Teams Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) (team2, d,1) team1 key = 3 Colliding Teams (team1, team2)

27 Option 2: Find New Paths for team 2 a b d f s 2 g g 2 1 s 2 1 c e s 1 1 g 1 1 a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out team1 {c,d,e} team2 {a,a,b,d} {b,b,d,f} Constraints (team2,d,1) a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out 3 in 3 out team1 team2

28 Storing Paths and Key Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) team1 key = 3 Colliding Teams (team1, team2) team2 key = 3

29 Looking for Collisions in Paths a b d f s 2 g g 2 1 s 2 1 c e s 1 1 g 1 1 a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out team1 {c,d,e} team2 {a,a,b,d} {b,b,d,f} Constraints (team2,d,1) a b c d e f s 2 1 s 2 2 s 1 1 g 2 2 g 1 1 g out 1 in 1 out 2 in 2 out 3 in 3 out team1 team2

30 Storing Colliding Teams Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) team1 key = 3 Colliding Teams (team1, team2) team2 key = 3 Colliding Teams None

31 Poping a Tree Node Ties are broken to favor nodes with fewest colliding pairs. Root key = 2 Colliding Teams (team1, team2) Earliest Collision (team1, team2, d,1) (team2, d,1) team1 key = 3 Colliding Teams (team1, team2) team2 key = 3 Colliding Teams None BINGO!

32 Edge Weights Reducing Possible Collisions Idea: Choose paths that have fewest collisions with other teams, when finding paths for a single team. Take into account the paths of other teams. Bias the search using a min-cost max-flow algorithm that finds a max flow with minimal total edge weights.

33 Edge Weights are Crucial Setups: neighbor grids with 10% randomly blocked cells. 5 agents per team. 5-minute time limits. CBM Unweighted CBM agents time success time success

34 Guarantees CBM is optimal and complete.

35 Comparisons Setups: neighbor grids with 10% randomly blocked cells. 5-minute time limits.

36 CBM is Faster CBM: Specialized solver. versus ILP (Integer Linear Program): Useful tool and easy to model. CBM ILP agents time success time (over solved instances) success

37 Spectrum: Anonymous Non-Anonymous Fixed 100 agents in total, 2 to 50 teams. Makespan Makespan Time Number of Teams Time [2 teams, 50 agents per team] [50 teams, 2 agents per team] Anonymous Non-Anonymous P NP-hard

38 Scalibility: Simulated Warehouse System Each instance has 420 agents: 210 incoming and 210 outgoing. CBM solves 40 out of the 50 Kiva instances within a time limit of 5 minutes each. Average running time over solved instances is seconds.

39 Takeaways TAPF: A mix of non-anonymous MAPF and anonymous MAPF. CBM: Guarantees optimality and completeness. Non-Anonymous MAPF NP-Hard Anonymous MAPF P + = TAPF