Effective Approaches for Partial Satisfaction (Over-subscription) Planning

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Effective Approaches for Partial Satisfaction
(Over-subscription) Planning

• Romeo Sanchez
• Menkes van den Briel
• Subbarao Kambhampati

Department of Computer Science and
Engineering Department of Industrial
Engineering Arizona State University Tempe,
Arizona
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Outline

• Background
• Example
• Approaches
• Optiplan
• Altaltps
• Sapaps
• Planning graph heuristics
• Results

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Background
In one day achieve the following 100 goals
RockData at WP 1, high-res pics at WP 2 3, .,
SoilData at WP 100
For all your demands, you couldve bought me a
better flash memory stick at least!
Given Actions with costs, and goals with
utilities, find a plan that has a highest
utility cost
No way I can achieve that many goals in one day
Its hard but here is the best I can do Goal1,
Goal5, Goal99

• Previous Approaches
• Highest utility goal first
• Estimating the set of most beneficial goals

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Background

• Complete satisfaction (traditional) planning
• Goal state G is a list of conjunctions G g1 ?
  g2 ? ? gn
• A plan that achieves n 1 goal fluents is as
  good as a plan that achieves 0 goal fluents
• Partial satisfaction planning (PSP)
• Goal state G is a list of fluents G g1, g2 ,
  , gn
• Goal fluents might have utilities, actions might
  have costs, therefore achieving a partial plan
  might be more beneficial than the null plan.
• Achieving all goal fluents might be impossible
• The goal state G may contain logically
  conflicting fluents
• There might not be enough resources to achieve
  all fluents in G

(goal (and (pointing satellite1 moon) (pointing
satellite1 mars) ))
(goal (and (have_rock rover1 waypoint1)
(have_rock rover1 waypoint2) ))
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PSP problems

• PSP Net benefit
• Given a planning problem P (F, A, I, G), and
  for each action a cost ca ? 0, and for each
  goal fluent f ? G a utility uf ? 0, and a
  positive number k. Is there a finite sequence of
  actions ? (a1, a2, , an) that starting from I
  leads to a state S that has net benefit ?f?(S?G)
  uf ?a?? ca ? k.

PLAN EXISTENCE
PLAN LENGTH
PSP GOAL
PSP GOAL LENGTH
PLAN COST
PSP UTILITY
PSP NET BENEFIT
PSP UTILITY COST
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Example

• Getting from Las Vegas (LV) to San Jose (SJ)

C action cost U(G) utility of goal
G G1,G2,G3,G4 goals P travel(LV,DL),
travel(DL,SJ), travel(SJ,SF) achieves G1, G2, G3
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Approaches

• Optiplan
• Integer programming based STRIPS planner
• Solves the PSP problem by encoding it as an
  integer program
• Altaltps
• Heuristic regression planner
• Solves the PSP problem through a goal selection
  heuristic
• Sapaps
• Heuristic forward state space planner
• Solves the PSP problem using an anytime A
  algorithm

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Optiplan

• Optiplan planning system
• Combines Graphplan (Blum Furst, 1995) with
  State Change Encoding (Vossen et al., 1999)
• As in the Blackbox planning system, Graphplan
  reduces the encoding size generated by Optiplan
• Computes optimal plans for a given parallel
  length
• Objective
• ?f?G Uf (x_addf,n x_preaddf,n x_maintainf,n)
  ? l?L ?a?A Ca ya,l
• Sum of goal utilities Sum of action
  cost

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Optiplan and partial satisfaction

• Objective
• 0 / Minimize actions
• Constraints
• Fluent changes
• Satisfy initial state
• Satisfy goal
• Fluent implications
• Action implications
• Total satisfaction planning goal satisfaction is
  treated as a hard constraint

• Objective
• Maximize net benefit
• Goal utility action cost
• Constraints
• Fluent changes
• Satisfy initial state
• Fluent implications
• Actions implications
• Partial satisfaction planning goal satisfaction
  is treated as a soft constraint

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Graphplan based cost propagation
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AltAltps

• AltAlt planning system
• Heuristic state-space search planner (Nguyen,
  Kambhampati Sanchez, 2002)
• Combines Graphplan (Blum Furst, 1995) with
  heuristic state-space search techniques (Bonet,
  Loerincs Geffner, 1997 Bonet Geffner, 1999
  McDermott 1999)
• AltAltps planning system
• Total enumeration on 2n goal subsets is too
  costly
• Selects a promising subset of the top-level goals
  upfront
• Searches for a plan using a regression state
  space search combined with cost-sensitive
  planning graph heuristics.

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AltAltps cost propagation

• Using a planning graph structure
• Propositions in the initial state come for free
  (they have zero cost)
• Other propositions have costs computed as
  follows
• Propagation procedures
• Max-propagation
• Sum-propagation

0
0
0
hl(p) Cost of proposition p at level l
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5
5
5
0
0
0
0 if p ? I hl(p) minhl-1(p), cost(a)
Cl(a) if l gt 0 ? otherwise
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8
4
4
4
4
l0
l1
l2
Cl(a) maxhl-1(q) q ? prec(a)
Cl(a) ?q ? prec(a) hl-1(q)
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AltAltps goal set selection

• Main idea
• Start with the original goal set G and an empty
  goal set G
• Iteratively add goals to G as long as the
  estimated NET BENEFIT increases
• The cost of adding another goal g to G depends
  on the goals that are already in G

G ? g
G
Cost for achieving G
Relaxed plan for G (Rp)
Residual cost for g
Rp for G ? g biased to re-use actions in Rp
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AltAltps cost-sensitive relaxed plan heuristic

• General procedure
• States are ranked during search using the relaxed
  plan heuristic and the propagated costs
• The idea is to compute the cost of a relaxed plan
  Rp in terms of the costs of the actions composing
  it.
• Heuristic value for S equal h(S) ?a?Rpcost(a)

1. Given a state S, remove the (sub)goal g from S
   that has highest hl(g)
2. Select the action that supports g with lowest
   cost (cost(a) Cl(a))
3. Regress S over a to get S S ? prec(a) \
   eff(a)
4. Stop when each proposition q? S is present in
   the initial state

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Sapaps
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SAPAPS a forward A approach for PSP
Anytime A Algorithm Search through best
beneficial nodes
A5 SampleRock
A1 Navigate(X,Y)
A2 SampleSoil(Y)
A4 Navigate(Y,Z)
A3 TakePicture

• Nodes evaluation
• g(S) U(S) C(S)
• h(S) U(RP(S)) C(RP(S))
• Beneficial Node
• g(S) gt 0 or U(S) gt C(S)
• Termination Node
• V S g(S) gt f(S)

g(S) Util(HasSoilData) Cost(A1,A2) h(S)
Util(Apply(A3,S)) Cost(A3)
A f(S) g(S) h(S)
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SAPAPS heuristic

• Heuristic Variation of SAPAs Approach
• Heuristically extracting the least cost relaxed
  plan using cost-function
• Remove unbeneficial goals and related actions

G1 G2 G3
A1
G1 G2
A1
A3
?
A3
A2
A4
C(A1) C(A2) gt U(G3)
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Empirical results
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Empirical results
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Future work