Performance of Coordinating Concurrent Hierarchical Planning
Agents Using Summary Information
Brad Clement and Ed Durfee
University of Michigan
Artificial Intelligence Laboratory
Overview
• Background– What is concurrent hierarchical plan coordination?
– What is summary information?
• Claims– Coordinating at abstract levels is much easier than coordinating at detailed
levels in finding some solution. (complexity analysis)
– Coordinating at abstract levels is better at finding optimal solutions.• search techniques and heuristics that leverage summary information• preliminary experimental results
• Other results– CHiP coordination algorithm is sound and complete.
– Resolving threats in a partial order plan is NP-complete.
Multi-level CoordinationA
B
DA
DB
Multi-level CoordinationA
B
DA
DB
A
B
DA
DB
Multi-level CoordinationA
B
DA
DB
A
B
DA
DB
A
B
DA
DB
Multi-level CoordinationA
B
DA
DB
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
Multi-level Coordination
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
Multi-level Coordination
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
A
B
DA
DB
temporalconstraints
blocked
Coordinating at Abstract Levels• Resolve conflicts at high level to minimize search time
• Better solutions may exist at lower levels
coordinationlevels
crispercrispersolutionssolutions
lowerlowercoordinationcoordination
costcost
flexibilityflexibility
Concurrent Hierarchical Plans (CHiPs)and Summary Information
• pre, in, & postconditions - sets of literals over a set of propositions
• summary information– external preconditions at(A, 0, 0)
– external postconditions at(A, 0, 4)
– internal conditions at(A, 1, 1)
– must, may, always, sometimes
at(A, 1, 2) must sometimes hold
at(A, 0, 1) may sometimes hold
havePower(A) must always hold
B - before
B B
B B B B B
A
B
DA
DB
0
1
2
0 1 2 3 4
Summary Information
• Summarize conditions of potential refinements at abstract levels
• Reason about abstract plan interactions among agents– resolve all conflicts at abstract level
– prune inconsistent refinement choices at abstract levels
– make refinement choices based on task interactions
Concurrent Hierarchical Plan Coordination
• Agents individually derive summary information for their plan hierarchies
• Coordinator requests summary information for expansions of agents’ hierarchies from the top down
• After each expansion, try to resolve threats by adding ordering constraints
• Algorithm shown to be sound and complete
Search for Coordinated Plan
• search state– set of expanded plans – set of blocked subplans– set of temporal constraints
• search operators– expand– block– constrain
blocked
blocked
temporal constraints
Reasoning at Abstract Levels Can Improve Performance
Total Cost
mid-level best
top-level best
primitive-level best
level computationtime
executiontime
top 4 60mid 159 40primitive 2375 35
A
B
DA
DB
Computation CostExecution Cost
Easier to Coordinate at Higher Levels
Number of summary conditions per plan grows exponentially up the hierarchyO(bd-ic)
b - branching factori - leveld - depthc - conditions per plan
Easier to Coordinate at Higher Levels
Number of summary conditions per plan grows exponentially up the hierarchyO(bd-ic)
Number of plans per level grows exponentially down the hierarchyO(bi)
b - branching factori - leveld - depthc - conditions per plan
Easier to Coordinate at Higher Levels
Complexity of identifying threats among plans is O(n2c´2) for n plan steps and c´ summary conditions per step orO(b2dc2)
b - branching factori - leveld - depthc - conditions per plan
Easier to Coordinate at Higher Levels
The number of orderings to test grows doubly exponentially down the hierarchyO(bi!)
b - branching factori - leveld - depthc - conditions per plan
Easier to Coordinate at Higher Levels
b - branching factori - leveld - depthc - conditions per plan
Resolving threats for a partial order plan is NP-complete (reduced from Hamiltonian Path)
Reasoning at Abstract Levels Can Improve Performance
Total Cost
mid-level best
top-level best
primitive-level best
level computationtime
executiontime
top 4 60mid 159 40primitive 2375 35
A
B
DA
DB
Computation CostExecution Cost
Search Techniques
• Prune inconsistent global plans • Branch & bound - abstract solutions
help prune space where cost is higher• “Expand most threats first” (EMTF)
– expand subplan involved in most threats
– focuses search on driving down to source of conflict
• “Fewest threats first” (FTF)– search plan states with fewest threats first
– or subplans involved in most threats are blocked first
NEO Domain Experiments
• Compare FAF’s and our strategies for ordering search states and ordering expansions
• 4 - 8 locations• 2 & 3 transports• no, partial, & complete overlap in locations visited
evacuateevacuate
noswitch
oneswitch
twoswitches
noswitch
oneswitch
twoswitches
cw ccw
go tofarthest
switch & goto farthest
go tosafe loc
move move movemove move movemove move
NEO Domain Experiments
Summary Information vs. FAF
1
10
100
1000
10000
100000
1000000
10000000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Problems
CP
U T
ime
FAFSummary Information
CPU Time in units of 1/100 CPU sec.
FAF only found solutions for 6 problems
• FTF-EMTF found solutions for 23 problems, 14 optimal
• FTF-ExCon found solutions for 19 problems, 12 optimal
• FAF-FAF found solutions for 22 problems, 14 optimal
• DFS-ExCon found solutions for 6 problems, 3 optimal (not shown)
Future Work
• What properties of plan hierarchies benefit which heuristics?
• For different domains, how can the hierarchies be restructured to take advantage of different heuristics?
• How can greater numbers of agents be continually coordinated as they accomplish, change, or add plans/goals?
Contributions• Sound and complete concurrent hierarchical plan
coordination algorithm
• Complexity analysis showing that resolving conflicts at higher levels is much easier than at lower levels
• Search techniques including FTF and EMTF heuristics that take advantage of summary information
• Preliminary experiments showing that these techniques can greatly improve the search for optimal plans