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

Topological Phenomena Local 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

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)