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

Jőrg Hoffmann

Alberts-Ludwigs-University Freiburg

Overview

• The Planning Benchmarks

• A Local Search Approach

• Local Search Topology

• Conclusion

Overview

• The Planning Benchmarks

• A Local Search Approach– FF Algorithms– AIPS´00 Competition

• Local Search Topology

• Conclusion

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

Overview

• The Planning Benchmarks

• A Local Search Approach

• Local Search Topology

• Conclusion

„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

„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

„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

„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

Overview

• The Planning Benchmarks

• A Local Search Approach– FF Algorithms– AIPS´00 Competition

• Local Search Topology

• Conclusion

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)

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]

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´) < h(S)

• Intuition: hill-climbing needs more „motion force“ towards the goal

FF Algorithms - Pruning

• Observation: often, GRAPHPLAN´s 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

Overview

• The Planning Benchmarks

• A Local Search Approach– FF Algorithms– AIPS´00 Competition

• Local Search Topology

• Conclusion

AIPS´00 Competition

• Planning systems competition alongside AIPS´00 [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

AIPS´00 - Logistics

AIPS´00 - Blocksworld(-arm)

AIPS´00 - Schedule

AIPS´00 - Freecell

AIPS´00 - Miconic-ADL

AIPS´00 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

FF vs. IPP in Gripper

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

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 FF´s approximative h+

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

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. FF‘s approximation of h+) – measure parameters of the resulting local search

topology (definitions adapted from [Frank et al.1997])

Topological PhenomenaDead ends

Measured: how many are there? Recognized? (i.e. h+ = ∞)?

Topological PhenomenaLocal Minima

Measured (amongst other things): how many are there?

Topological Phenomena

Benches

Measured (amongst other things): maximal exit distance

h+ Topology in Small Instances

In lowermost class, enforced hill-climbing is polynomial!FF approximation similar: some, but few local minima

A Visualized Example: Gripper

A Visualized Example: Hanoi

A Visualized Example: Hanoi

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

The proven Topology of h+

Reasons for h+ Topology

• Invertible actions: actions a to which there exists an inverse action undoing exactly a‘s 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

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+

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

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

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

Logistics: Local Minima

Logistics: Maximal Exit Distance

Overview

• The Planning Benchmarks

• A Local Search Approach

• Local Search Topology

• Conclusion

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

Conclusion - Local Search

It is certainly an extreme example, but nevertheless:

Utilizing problem structure can be crucial

Thanks to: Bernhard Nebel; Jana Koehler;

for doing successful local search(though you‘d normally first identify the structure, then try to utilize it)