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 Jőrg 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 • AIPS´00 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

6. „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

7. „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

8. „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

9. „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

10. Overview • The Planning Benchmarks • A Local Search Approach • FF Algorithms • AIPS´00 Competition • Local Search Topology • Conclusion

11. 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)

12. 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]

13. 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

14. 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 thoseselected by the first step of the relaxed plan for S

15. Overview • The Planning Benchmarks • A Local Search Approach • FF Algorithms • AIPS´00 Competition • Local Search Topology • Conclusion

16. 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

17. AIPS´00 - Logistics

18. AIPS´00 - Blocksworld(-arm)

19. AIPS´00 - Schedule

20. AIPS´00 - Freecell

21. AIPS´00 - Miconic-ADL

22. 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

23. FF vs. IPP in Gripper

24. 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

25. 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+

26. 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

27. 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])

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

29. Topological Phenomena Local Minima Measured (amongst other things): how many are there?

30. Topological Phenomena Benches Measured (amongst other things): maximal exit distance

31. h+ Topology in Small Instances In lowermost class, enforced hill-climbing is polynomial! FF approximation similar: some, but few local minima

32. A Visualized Example: Gripper

33. A Visualized Example: Hanoi

34. A Visualized Example: Hanoi

35. 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

36. The proven Topology of h+

37. 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

38. Reasons for h+ Topology • Actions that are respected by the relaxation: if astarts 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+

39. 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

40. 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+: • checkmaximum over exit distances in large instances

41. 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

42. Logistics: Local Minima

43. Logistics: Maximal Exit Distance

44. Overview • The Planning Benchmarks • A Local Search Approach • Local Search Topology • Conclusion

45. 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

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