Finding Admissible Bounds for Over-subscribed Planning Problems

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Finding Admissible Bounds for Over-subscribed
Planning Problems

• J. Benton

Menkes van den Briel
Subbarao Kambhampati
Arizona State University
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Is this plan good?
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How good is a given plan
How to drive a planner to find a good plan
Related

e.g., when we have many soft goals
Especially important when quality may vary widely
Admissible heuristics
Helps per-node use
Helps one-shot use
Need a heuristic schema that admits degrees of
relaxation
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(No Transcript)
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Challenges
1. Build a strong admissible heuristic 2.
Provide a way to add relaxation for varied use
An integer programming (IP) based heuristic
Use the linear programming (LP) relaxation
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PSPUDPartial Satisfaction Planning with Utility
Dependency
Actions have cost
Goal sets have utility
loc1
loc2
(at t loc2) (in p1 t)
(at t loc1) (in p1 t)
(at t loc1) (at p1 loc2)
(at t loc2) (at p1 loc2)
S3
S0
S1
S2
(move t loc2)
(unload p1 loc2)
(move t loc1)
cost 20
cost 20
cost 5
sum cost 20
sum cost 25
sum cost 45
util(S1) 0
util(S3) 10106080
util(S0) 10
util(S2) 10
net benefit(S0) 10-010
net benefit(S1) 0-20-20
net benefit(S2) 10-25-15
net benefit(S3) 80-4535
utility((at t loc1) (at p1 t)) 60
utility((at t loc1)) 10
utility((at p1 loc2)) 10
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Building a Heuristic
A network flow model on variable transitions
Capture relevant transitions with multi-valued
fluents
add initial states
add prevail constraints
add goal states
add cost on actions
add utility on goals
loc1
loc2
package
truck
util 10
util 10
cost 5
util 60
cost 20
cost 5
cost 20
cost 5
cost 5
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Building a Heuristic
Constraints of this model
1. If an action executes, then all of its effects
and prevail conditions must also.
2. If a fact is deleted, then it must be added to
re-achieve a value.
3. If a prevail condition is required, then it
must be achieved.
4. A goal utility dependency is achieved if its
goals are achieved.
package
truck
util 10
util 10
cost 5
util 60
cost 20
cost 5
cost 20
cost 5
cost 5
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Formulation
Variables
action(a) ? Z The number of times a ? A is executed
effect(a,v,e) ? Z The number of times a transition e in state variable v is caused by action a
prevail(a,v,f) ? Z The number of times a prevail condition f in state variable v is required by action a
endvalue(v,f) ? 0,1 Equal to 1 if value f is the end value in a state variable v
goaldep(k) Equal to 1 if a goal dependency is achieved
Parameters
cost(a) the cost of executing action a ? A
utility(v,f) the utility of achieving value f in state variable v
utility(k) the utility of achieving achieving goal dependency Gk
1. If an action executes, then all of its effects
and prevail conditions must also.
action(a) Seffects of a in v effect(a,v,e)
Sprevails of a in v prevail(a,v,f)
2. If a fact is deleted, then it must be added to
re-achieve a value.
1if f ? s0v Seffects that add f
effect(a,v,e) Seffects that delete f
effect(a,v,e) endvalue(v,f)
3. If a prevail condition is required, then it
must be achieved.
1if f ? s0v Seffects that add f
effect(a,v,e) prevail(a,v,f) / M
4. A goal utility dependency is achieved if its
goals are achieved.
goaldep(k) Sf in dependency k endvalue(v,f)
Gk 1 goaldep(k) endvalue(v,f) ? f in
dependency k
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Formulation
Variables
action(a) ? Z The number of times a ? A is executed
effect(a,v,e) ? Z The number of times a transition e in state variable v is caused by action a
prevail(a,v,f) ? Z The number of times a prevail condition f in state variable v is required by action a
endvalue(v,f) ? 0,1 Equal to 1 if value f is the end value in a state variable v
goaldep(k) Equal to 1 if a goal dependency is achieved
Parameters
cost(a) the cost of executing action a ? A
utility(v,f) the utility of achieving value f in state variable v
utility(k) the utility of achieving achieving goal dependency Gk
Objective Function
Sv?V,f?Dv utility(v,f) endvalue(v,f) Sk?K
utility(k) goaldep(k) Sa?A cost(a) action(a)
Maximize Net Benefit
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Experimental Setup
Three modified IPC 3 domains zenotravel,
satellite, rovers
(maximize net benefit)
One IPC 5 domain Rovers, simple preferences
(minimize (goal achievement violations action
cost))
Compared with
, a cost propagation-based heuristic
heuristic value at initial state versus optimal
plan
Found using a branch and bound search
LP gt IP gt OPTIMAL
maximizing
LP lt IP lt OPTIMAL
minimizing
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Results
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Results
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Results
IP
LP
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Summary

• IP gives bound on quality of plan
• Doubly relaxed (LP) to provide heuristic for
  search (Search I Session Monday at 410 pm)

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Future Work

• Improve encoding (to give better LP values)
• Use fluent merging