1. Performance of Coordinating Concurrent Hierarchical Planning Agents Using Summary Information
Brad Clement and Ed Durfee
University of Michigan
Artificial Intelligence Laboratory

2. Overview Background Claims Other results
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.

3. Multi-level CoordinationDA B DB

4. Multi-level CoordinationDA DA A A B DB B DB

5. Multi-level CoordinationDA DA A B DB B DB DA A B DB

6. Multi-level CoordinationDA DA A B DB B DB DA DA A A DB B DB B

7. Multi-level CoordinationDA DA A A B DB B DB DA DA DA A A A DB B DB B B DB

8. Multi-level CoordinationDA DA A A B DB B DB DA DA DA A A A DB B DB B B DB
blocked
temporal constraints

9. Coordinating at Abstract Levels
Resolve conflicts at high level to minimize search time
Better solutions may exist at lower levels
crisper
solutions
lower coordination cost
coordination
levels
flexibility

10. 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 A B DA DB 1 2 3 4 B B
B - before B B B B B

11. 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

12. 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

13. Search for Coordinated Plan
search state
set of expanded plans
set of blocked subplans
set of temporal constraints
search operators
expand
block
constrain
blocked
temporal constraints
blocked

14. Reasoning at Abstract Levels Can Improve PerformanceDA DB
Total Cost
top-level best
mid-level best
primitive-level best
Computation Cost Execution Cost

15. Easier to Coordinate at Higher Levels
b - branching
factor
i - level
d - depth
c - conditions per plan
Number of summary conditions per plan grows exponentially up the hierarchy O(bd-ic)

16. Easier to Coordinate at Higher Levels
b - branching
factor
i - level
d - depth
c - conditions per plan
Number of summary conditions per plan grows exponentially up the hierarchy O(bd-ic)
Number of plans per level grows exponentially down the hierarchy O(bi)

17. Easier to Coordinate at Higher Levels
b - branching
factor
i - level
d - depth
c - conditions per plan
Complexity of identifying threats among plans is O(n2c´2) for n plan steps and c´ summary conditions per step or O(b2dc2)

18. Easier to Coordinate at Higher Levels
b - branching
factor
i - level
d - depth
c - conditions per plan
The number of orderings to test grows doubly exponentially down the hierarchy
O(bi!)

19. Easier to Coordinate at Higher Levels
b - branching
factor
i - level
d - depth
c - conditions per plan
Resolving threats for a partial order plan is NP-complete (reduced from Hamiltonian Path)

20. Reasoning at Abstract Levels Can Improve PerformanceDA DB
Total Cost
top-level best
mid-level best
primitive-level best
Computation Cost Execution Cost

21. 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

22. 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

23. NEO Domain Experiments
evacuate
evacuate
no switch
one switch
two switches
no switch
one switch
two switches cw
ccw
go to farthest
switch & go to farthest
go to
safe loc
move
move
move
move
move
move
move
move

24. Summary Information vs. FAF
CPU Time in units of 1/100 CPU sec.
FAF only found solutions for 6 problems

25. 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)

26. 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?

27. 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