Utilizing Problem Structure in Local Search: The Planning Benchmarks as a Case Study

1
Utilizing Problem Structure in Local Search The
Planning Benchmarks as a Case Study

• Jorg Hoffmann
• Alberts-Ludwigs-University Freiburg

2
Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Conclusion

3
Overview

• The Planning Benchmarks
• A Local Search Approach
• FF Algorithms
• AIPS00 Competition
• Local Search Topology
• Conclusion

4
Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Gathering Insights Looking at Small Instances
• The Topology of h
• The Topology of Approximating h
• Conclusion

5
Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Conclusion

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The Planning Benchmarks

• Assembly, Blocksworld-arm, Blocksworld-no-arm,
  Briefcaseworld,Ferry, Fridge, Freecell, Grid,
  Gripper, Hanoi, Logistics, Miconic-ADL,
  Miconic-SIMPLE, Miconic-STRIPS, Movie, Mprime,
  Mystery, Schedule, Simple-Tsp, Tyreworld

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The Planning Benchmarks

• Assembly, Blocksworld-arm, Blocksworld-no-arm,
  Briefcaseworld,Ferry, Fridge, Freecell, Grid,
  Gripper, Hanoi, Logistics, Miconic-ADL,
  Miconic-SIMPLE, Miconic-STRIPS, Movie, Mprime,
  Mystery, Schedule, Simple-Tsp, Tyreworld

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The Planning Benchmarks

• Assembly, Blocksworld-arm, Blocksworld-no-arm,
  Briefcaseworld,Ferry, Fridge, Freecell, Grid,
  Gripper, Hanoi, Logistics, Miconic-ADL,
  Miconic-SIMPLE, Miconic-STRIPS, Movie, Mprime,
  Mystery, Schedule, Simple-Tsp, Tyreworld

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The Planning Benchmarks

• Assembly, Blocksworld-arm, Blocksworld-no-arm,
  Briefcaseworld,Ferry, Fridge, Freecell, Grid,
  Gripper, Hanoi, Logistics, Miconic-ADL,
  Miconic-SIMPLE, Miconic-STRIPS, Movie, Mprime,
  Mystery, Schedule, Simple-Tsp, Tyreworld

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Overview

• The Planning Benchmarks
• A Local Search Approach
• FF Algorithms
• AIPS00 Competition
• Local Search Topology
• Conclusion

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FF Algorithms

• FF can be seen as a refinement of HSP 1.0
• search forward in the state space
• relax planning task by ignoring delete lists
• Main Differences Hoffmann Nebel 2001
• heuristic (different
  approximation of h)
• search strategy (different hill-climbing
  variant)
• pruning technique (new)

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FF Algorithms - Heuristic

• Approach often used in heuristic search relax
  problem, solve relaxation
• In planning ignore delete lists Bonet et
  al.1997
• Optimal relaxed solution length h admissible but
  NP-hard to compute Bylander 1994
• HSP 1.0 approximate h by weight value sums
• FF approximate h by running a relaxed version
  of GRAPHPLAN Blum Furst 1997

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FF Algorithms - Search

• Local search as state evaluation is costly
• HSP 1.0 (standard) hill-climbing
• FF enforced hill-climbing
• start in initial state
• in a state S, do breadth first search for S such
  that h(S) lt h(S)
• Intuition hill-climbing needs more motion
  force towards the goal

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FF Algorithms - Pruning

• Observation often, GRAPHPLANs relaxed solutions
  are close to what needs to be done, at least in
  first step
• in Gripper, for example, actions that drop balls
  into room A are never selected
• Restrict action choice in any state S to those
  selected by the first step of the relaxed plan
  for S

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Overview

• The Planning Benchmarks
• A Local Search Approach
• FF Algorithms
• AIPS00 Competition
• Local Search Topology
• Conclusion

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AIPS00 Competition

• Planning systems competition alongside AIPS00
  Bacchus 2001
• 15 participants, 12 in fully automated track
• 5 domains, around 50 - 200 scaling instances each
• we briefly look at the runtime curves in the
  fully automated track

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AIPS00 - Logistics
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AIPS00 - Blocksworld(-arm)
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AIPS00 - Schedule
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AIPS00 - Freecell
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AIPS00 - Miconic-ADL
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AIPS00 Competition

• As a result of the competition, FF
• was nominated Group A Distinguished Performance
  Planning System (together with TalPlanner from
  the hand-tailored track)
• won the Schindler Award for Best Performance in
  the Miconic domain, ADL track
• Note we have only briefly seen one part of the
  competition

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FF vs. IPP in Gripper
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Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Gathering Insights Looking at Small Instances
• The Topology of h
• The Topology of Approximating h
• Conclusion

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Local Search Topology

• The behaviour of local search depends crucially
  on the topology of the search space (studied in
  SAT, e.g. Frank et al. 1997)
• Identify, following Frank et al. 1997, the
  topology of the benchmarks, under h and FFs
  approximative h

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Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Gathering Insights Looking at Small Instances
• The Topology of h
• The Topology of Approximating h
• Conclusion

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Gathering Insights

• Start by looking at small instances Hoffmann
  2001
• in the 20 domains, randomly generate suits of
  small examples
• build the state spaces and compute h to all
  states (resp. FFs approximation of h)
• measure parameters of the resulting local search
  topology (definitions adapted from Frank et
  al.1997)

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Topological Phenomena
Dead ends
Measured how many are there? Recognized? (i.e.
h 8)?
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Topological Phenomena
Local Minima
Measured (amongst other things) how many are
there?
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Topological Phenomena
Benches
Measured (amongst other things) maximal exit
distance
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h Topology in Small Instances
In lowermost class, enforced hill-climbing is
polynomial! FF approximation similar some, but
few local minima
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A Visualized Example Gripper
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A Visualized Example Hanoi
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A Visualized Example Hanoi
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Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Gathering Insights Looking at Small Instances
• The Topology of h
• The Topology of Approximating h
• Conclusion

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The proven Topology of h
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Reasons for h Topology

• Invertible actions actions a to which there
  exists an inverse action undoing exactly as
  effects
• Example Logistics
• load obj truck --- unload obj truck
• drive loc1 loc2 --- drive loc2 loc1
• Implies non-existence of dead ends, and of local
  minima with see next slide

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Reasons for h Topology

• Actions that are respected by the relaxation if
  a starts an optimal plan from S, then a also
  starts an optimal relaxed plan from S
• Example Logistics
• load obj truck obj must be transported, and
  there is no other way of doing that
• drive loc1 loc2 some obj must be loaded/unloaded
  at loc2, again no other choice for the relaxed
  plan
• If all actions are invertible and respected by
  the relaxation, then there are no local minima
  under h

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Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Gathering Insights Looking at Small Instances
• The Topology of h
• The Topology of Approximating h
• Conclusion

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The Topology of Approximating h

• Dead ends behave provably the same
• In domains where no local minima exist under h
• check local minima percentage under approximative
  (FF) heuristic in large instances
• In domains where maximal exit distance constant
  under h
• check maximum over exit distances in large
  instances

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Investigating Large Instances

• Take Samples from State Spaces (following
  Frank et al. 1997)
• randomly generate suits of large instances
• repeatedly, walk a random number of random steps
  into the state space, ending in a state S
• check whether S lies on a local minimum, and what
  the exit distance is
• visualize data against generator parameters

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Logistics Local Minima
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Logistics Maximal Exit Distance
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Overview

• The Planning Benchmarks
• A Local Search Approach
• Local Search Topology
• Conclusion

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Conclusion - Planning

• Critically time to move on to other benchmarks?
• agree time and resources
• disagree only NP-hard problems for benchmarking
• Positively we have a good suboptimal planner!
• we know where it works well
• we know why it works well

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Conclusion - Local Search
It is certainly an extreme example, but
nevertheless
Utilizing problem structure can be crucial
for doing successful local search
(though youd normally first identify the
structure, then try to utilize it)
Thanks to Bernhard Nebel Jana Koehler