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Integer Programming Approaches for Automated Planning

Integer Programming Approaches for Automated Planning. Menkes van den Briel Department of Industrial Engineering Arizona State University menkes@asu.edu http://www.public.asu.edu/~dbvan1/. What is automated planning?. Ordering problem

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Integer Programming Approaches for Automated Planning

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  1. Integer Programming Approaches for Automated Planning Menkes van den BrielDepartment of Industrial EngineeringArizona State Universitymenkes@asu.eduhttp://www.public.asu.edu/~dbvan1/

  2. What is automated planning? Ordering problem Scheduling is the problem of deciding when to execute a set of actions NP-complete Selection and ordering problem Planning is deciding both what actions need to be done and when to execute them PSPACE-complete Scheduling Planning

  3. What is automated planning? • Creating a computer program to produce a plan, a sequence of actions that will transform the world from some given initial state to a desired goal state 1 2 1 2 Initial states0 S Goalg S PlanP = a1, …, an Action Actions are state transformation functions

  4. What is automated planning? • Creating a computer program to produce a plan, a sequence of actions that will transform the world from some given initial state to a desired goal state Initial states0 S Goalg S PlanP = a1, …, an Action si sj Actions are state transformation functions

  5. Planning applications • Autonomous vehicles • Mars rovers • Underwater robotics • Remote agent experiment • Games • Bridge Baron • General game playing • Others • Manufacturing process planning • Composition of web services • Cyber Security

  6. Planning by integer programming Operations research (OR) Scheduling problems typically involve solving hard optimization problems Integer programming (IP), branch-and-bound Artificial intelligence (AI) Planning problems typically involve solving hard feasibility problems Constraint satisfaction, satisfiability (SAT), A* search Scheduling Planning

  7. Planning by integer programming • Very little focus on integer programming approaches for planning • [Bylander, 1997] • [Bockmayr and Dimopoulos, 1998, 1999] • [Kautz and Walser, 1999] • [Vossen et al., 1999] • [Dimopoulos, 2001] • [Dimopoulos and Gerevini, 2002]

  8. Why this lack of interest? • IP-based approaches simply don’t work • “Lplan [a linear programming-based heuristic for optimal planning] was often slower than the other algorithms primarily due to the time to evaluate the linear programming heuristic”[Bylander, 1997] • SAT-based approaches are much faster • SAT-based planners have successfully participated in IPC1, IPC2, IPC4, and IPC5 • Traditionally there has been little focus on plan quality • Planning is PSPACE-complete, so finding a feasible plan is already hard enough

  9. Counter arguments • IP-based approaches do work • Optiplan, first IP-based planner to take part in the IPC series • Ranked 2nd in four out of seven domains in IPC4 in the optimal track for propositional domains • IP-based approaches can compete with SAT-based approaches • Represent planning as a set of interdependent network flow problems • Generalize the notion of action parallelism • Shift in focus towards optimal planning • Applied formulations to partial satisfaction planning problems • Developed a novel framework for optimal planning • Utilized LP relaxations in deriving quality sensitive heuristics

  10. Contributions • IP-based approaches do work • IP-based approaches can compete with SAT-based approaches • Shift in focus towards optimal planning • [Van den Briel, and Kambhampati. Journal of Artificial Intelligence Research, 2005] • [Van den Briel, Vossen, and Kambhampati. ICAPS, 2005] • [Van den Briel, Vossen, and Kambhampati. Journal of Artificial Intelligence Research, 2008] • [Van den Briel, et al. AAAI, 2004] • [Do, Benton, van den Briel, and Kambhampati. IJCAI, 2007] • [J. Benton, van den Briel, and Kambhampati. ICAPS, 2007] • [Van den Briel, Benton, Kambhampati, and Vossen. CP, 2007]

  11. 1. IP approaches do work • Optiplan • IP-based planner that extends the state change formulation by [Vossen et al., 1999] [van den Briel, and Kambhampati, 2005]

  12. Summary of results • International planning competition (IPC) • Bi-annual event • Provides data sets (domains) that are used as benchmarks • IPC4 • 7 competition domains • 7 participating planners in the “optimal” track • Domains • Pipesworld • Control the flow of oil derivatives through a pipeline network, obeying various constraints such as product compatibility and tankage restrictions • Satellite • Collect image data with a number of satellites • Philosophers, Optical telegraph • Involves finding deadlocks in communication protocols

  13. Summary of results

  14. 2. IP versus SAT approaches • Represent planning as a set of interdependent network flow problems • One network flow problem for each state variable in the planning domain • Nodes correspond to the values of the state variables, arcs correspond to the value transitions • Generalize the notion of action parallelism • Reduces the plan length of the solution plan (and thus the size of the formulation)

  15. Logistics example 1 2 P T Truck Load(P,T,1)Unload(P,T,1) 1 Drive(1,2) Drive(2,1) 2 Load(P,T,1)Unload(P,T,1) Package 1 Load(P,T,1) unload(P,T,1) 2 Load(P,T,2) unload(P,T,2) T States are described by state variables

  16. Logistics example 1 2 Prevail Truck Load(P,T,1)Unload(P,T,1) 1 Drive(1,2) Drive(2,1) 2 Load(P,T,1)Unload(P,T,1) Package 1 Load(P,T,1) unload(P,T,1) Effect 2 Load(P,T,2) unload(P,T,2) T Actions are state transformation functions

  17. One state change (1SC) • Network representation • Logistics example Prevail f f f Effect g g g h h h Plan step Truck 1 1 2 2 Planning involves considering plans of increasing length Package 1 1 2 2 t t t = 1

  18. One state change (1SC) • Network representation • Logistics example Prevail f f f Effect g g g h h h Load(P,T,1) Drive(1,2) Unload(P,T,2) Truck 1 1 1 1 2 2 2 2 Load(P,T,1) - Unload(P,T,2) Package 1 1 1 1 2 2 2 2 t t t t t = 1 t = 2 t = 3

  19. 1SC formulation • Constraints • State changes (network flow), for allc  CgCycf,g,t= 1{f  I} for f  DchCycg,h,t+1= fCycg,h,t for f  Dc , 1  t < T fCycf,g,T= 1 for g  G • Effect implications, for allc  C, 1  t  TaA:(f,g)SC(a)xa,t = ycf,g,tfor f, g  Dc, f  g xa,t  ycf,f,tfor a  A, f PR(a)

  20. Summary of results • Experimental setup • Domains from IPC2, IPC3 • Comparing 1SC formulation versus SATPLAN04 (winner of the “optimal“ track IPC4) • 2.67GHz CPU with 1.0GB memory • Domains • Logistics, Driverlog • Involves driving trucks (and flying airplanes) around to deliver packages between locations • Blocksworld • Stacking and unstacking towers of blocks • Zenotravel • Transporting people around in planes, using different modes of movement: fast and slow

  21. Summary of results

  22. 2. IP versus SAT approaches • Represent planning as a set of interdependent network flow problems • One network flow problem for each state variable in the planning domain • Nodes correspond to the values of the state variables, arcs correspond to the value transitions • Generalize the notion of action parallelism • Reduces the plan length of the solution plan (and thus the size of the formulation)

  23. Generalized one state change (G1SC) • Network representation • Example Prevail f f f Effect g g g h h h Load(P,T,1)Drive(1,2) Unload(P,T,2) Truck 1 1 1 2 2 2 Load(P,T,1) Unload(P,T,2) Package 1 1 1 2 2 2 t t t t = 1 t = 2

  24. Implied precedences (G1SC) • Example A4 A1 A3 A1,A2 A3 A4 A2 Implied precendence graph

  25. Implied precedences (G1SC) • Example • Ordering (cycle elimination) constraints ensure a feasible ordering of the actions A4 A1 A3 A1,A2 A3 A4 A2 Implied precendence graph A4 A1 xA1,t + xA3,t + xA4,t  2

  26. G1SC formulation • Constraints • State changes (network flow), for allc  CgCycf,g,t= 1{f  I} for f  DchCycg,h,t+1= fCycg,h,t for f  Dc, 1  t  T fCycf,g,T= 1 for g  G • Effect implications, for allc  C, 1 t  T aA:(f,f)SC(a)xa,t= ycf,g,t for f, g  Dc, f  g, xa,t  ycf,f,t + gDc:f≠g (ycg,f,t + ycf,g,t)for a  A, f PR(a) • Ordering (Cycle elimination) constraints  aV()xa,t  |V()| – 1 for all cycles G, 1  t  T

  27. Branch-and-cut START STOP Initialize LP no Nodes found? yes LP solver Feasible? no Fathom Node selection yes Z_lp < Z*? no yes Cut generation Cuts found? yes no Integer? no Branching yes

  28. State change path (PathSC) • Network representation • Example Prevail f f f Effect g g g h h h Load(P,T,1)Drive(1,2)Unload(P,T,2) Truck 1 1 2 2 load(P,T,1)unload(P,T,2) Package 1 1 2 2 t t t = 1

  29. Summary of results

  30. Summary of results [van den Briel, Vossen, and Kambhampati, 2005, 2008]

  31. 3. Shift towards optimal planning • Applied formulations to partial satisfaction planning problems • Developed a novel framework for optimal planning • Utilized LP relaxations in deriving quality sensitive heuristic search approaches

  32. Partial satisfaction planning • PLAN LENGTH is PSPACE-complete • [Bylander, 1994] • PSP UTILITY COST is PSPACE-complete • [Van den Briel, et al., 2004] Total Satisfaction Problems PSP UTILITY COST PSP NET BENEFIT PLAN COST PSP UTILITY PSP GOAL LENGTH Partial SatisfactionProblems PLAN LENGTH PSP GOAL PLAN EXISTENCE

  33. Framework for optimal planning • For step-based IP formulations optimality is restricted to the length of the plan Plan step Load(P,T,1) Drive(1,2) Unload(P,T,2) Truck 1 1 1 1 2 2 2 2 Load(P,T,1) - Unload(P,T,2) Package 1 1 1 1 2 2 2 2 t t t t t = 1 t = 2 t = 3

  34. Framework for optimal planning 1 2 P T Truck Load(P,T,1)Unload(P,T,1) 1 Drive(1,2) Drive(2,1) 2 Load(P,T,1)Unload(P,T,1) Package 1 Load(P,T,1) unload(P,T,1) 2 Load(P,T,2) unload(P,T,2) T

  35. Action selection formulation • Variables • xa Z+, for a  A; xa is equal to the number of times action a is executed • yv(c,a) Z+, for v  V, a  A, a  –(c); yv(c,a) is equal to the number of times transition v(c,a) is executed • Objective function • MIN aAcaxa • Constraints • av+(e)yv(c,a) – a v–(e)yv(c,a) • av+(e)yv(c,a) = xa No time indicesNo upper bounds 1 if c c0,v, c g–1 if c= c0,v, c g0 otherwise

  36. Concurrent automata • Given a set of state variables V = {v1, …, vn} • For each v V we define a deterministic automaton Gv = (Dv, Av, v, v, c0,v, gv) • Dv is a finite set of states corresponding to the domain of state variable v • Av is a finite set of actions associated with the transitions in Gv • v : Dv  A  Dv is the transition function • v : Dv  2A is the active action function • c0,v  S is the initial state of state variable v • gv  S is a set of goal states of state variable v

  37. Parallel composition • The parallel composition of the two automata G1 and G2 is the automaton G1||G2 := (D1D2, A1A2, 1||2, 1||2, (c0,1, c0,2), g1g2) • 1||2((c1,c2),a) := • 1||2(c1,c2) := [1(c1)2(c2)]  [1(c1)\A2][2(c2)\A1] (1(c1,a), 2(c2,a) if a  1(c1)2(c2)(1(c1,a), c2) if a  1(c1)\A2(c1,2(c2,a)) if a  2(c2)\A1undefined otherwise

  38. Logistics example 1 2 P T Truck Load(P,T,1)Unload(P,T,1) 1 Drive(1,2) Drive(2,1) 2 Load(P,T,1)Unload(P,T,1) Package 1 Load(P,T,1) unload(P,T,1) 2 Load(P,T,2) unload(P,T,2) T

  39. Simple logistics example 1 2 P T Truck|| Package 2,1 Drive(2,1) Drive(1,2) 1,2 1,1 Drive(2,1) Drive(1,2) Unload(P, T,1) Load(P, T,1) 2,2 1,T Unload(P, T,2) Drive(2,1) Load(P, T,2) Drive(1,2) 2,T

  40. Summary of results Highlighted values equal optimal solution

  41. Summary of results

  42. Utilize LP in heuristic search BBOP-LP planner [Benton, van den Briel, and Kambhampati, 2007]

  43. Summary • IP-based approaches do work • Optiplan, first IP-based planner to take part in the IPC series • Ranked 2nd in four out of seven domains in IPC4 in the optimal track for propositional domains • IP-based approaches can compete with SAT-based approaches • Represent planning as a set of interdependent network flow problems • Generalize the notion of action parallelism • Shift in focus towards optimal planning • Applied formulations to partial satisfaction planning problems • Developed a novel framework for optimal planning • Utilized LP relaxations in deriving quality sensitive heuristics

  44. Publications status • Journal • [M.H.L. van den Briel, and S. Kambhampati. Optiplan: Unifying IP-based and graph-based planning. Journal of Artificial Intelligence Research, 24:623-635, 2005] • [M.H.L van den Briel, T. Vossen, and S. Kambhampati. Loosely coupled formulation for automated planning: An integer programming perspective. Journal of Artificial Intelligence Research, 31:217-257, 2008] • [(In progress) M.H.L van den Briel, T. Vossen, S. Kambhampati and J. Fowler. Optimal automated planning] • Conference • [M.H.L. van den Briel, R. Sanchez, M.B. Do, and S. Kambhampati. Effective approaches for partial satisfaction (oversubscription) planning. In Proceedings of AAAI, pages 562-569, 2004] • [M.H.L. van den Briel, T. Vossen, and S. Kambhampati. Reviving integer programming approaches for AI planning: A branch-and-cut framework. In Proceedings of ICAPS, pages 161-170, 2005] • [M.B. Do, J. Benton, M.H.L. van den Briel, and S. Kambhampati. Planning with goal utility dependencies. In Proceedings of IJCAI, pages 1872-1878, 2007] • [J. Benton, M.H.L. van den Briel, and S. Kambhampati. A hybrid linear programming and relaxed plan heuristic for partial satisfaction planning problems. In Proceedings of ICAPS, pages 24-41, 2007] • [M.H.L. van den Briel, J. Benton, S. Kambhampati, and T. Vossen. An LP-based heuristic for optimal planning. In Proceedings of CP, pages 651-665, 2007] Cited by 6 Cited by 31 Cited by 15 Cited by 3 Cited by 4 Cited by 3

  45. Publications status • Workshop and posters • [M.H.L. van den Briel, R. Sanchez, and S. Kambhampati. Over-Subscription in Planning: a Partial Satisfaction Problem. In Proceedings of ICAPS Workshop on Integrating Planning into Scheduling, 2005] • [M.H.L. van den Briel,. Kambhampati, and T. Vossen. Planning with numerical state variables through mixed integer programming. In Proceedings of ICAPS Poster Session, pages 5-8, 2005] • [M.H.L. van den Briel,. Kambhampati, and T. Vossen. Planning with preferences and trajectory constraints by integer programming. In Proceedings of ICAPS Workshop on Preferences and Soft Constraints in Planning, pages 19-22, 2006] • [J. Benton, M.H.L. van den Briel,. Kambhampati. Finding admissible bounds for oversubscription planning problems. In Proceedings of ICAPS Workshop on Heuristics for Domain-Independent Planning: Progress, Ideas, Limitations, Challenges, 2007] • [M.H.L. van den Briel,. Kambhampati, and T. Vossen. Fluent merging: A general technique to improve reachability heuristics and factored planning. In Proceedings of ICAPS Workshop on Heuristics for Domain-Independent Planning: Progress, Ideas, Limitations, Challenges, 2007] Cited by 5 Cited by 1 Cited by 1 Citation count by Google Scholar

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