1 Integer Programming Approaches for Automated Planning Menkes van den Briel Department of Industrial Engineering Arizona State University [email protected].

1

Integer Programming Approaches for Automated Planning

Menkes van den BrielDepartment of Industrial Engineering

Arizona State [email protected]

http://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

Initial states0 S

Goalg S

PlanP = a1, …, an

Action

Actions are state transformation functions

1 2 1 2

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

Actionsi 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

West

North

East

South

62

8Q

QJ65

97

AK53

A9

West

North

East

South

62

8Q

QJ65

97

AK53

A9

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

1. 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]

2. SAT-based approaches are much faster– SAT-based planners have successfully participated in IPC1,

IPC2, IPC4, and IPC5

3. Traditionally there has been little focus on plan quality– Planning is PSPACE-complete, so finding a feasible plan is

already hard enough

Why this lack of interest?

9

1. 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

2. 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

3. 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

Counter arguments

10

1. IP-based approaches do work•

2. IP-based approaches can compete with SAT-based approaches•

3. Shift in focus towards optimal planning

Contributions

– [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

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10

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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Phillosophers

So

luti

on

tim

e (s

ec.)

Satplan04OptiplanCPTHSPS-ATP4

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

Optical telegraph

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Satplan04OptiplanCPTHSPS-ATP4

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Pipesworld(tankage)

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ec.)

Satplan04

Optiplan

CPT

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Satellite

So

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Satplan04

Optiplan

CPT

14

2. IP versus SAT approaches

1. 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

2. Generalize the notion of action parallelism – Reduces the plan length of the solution plan (and thus the size

of the formulation)

15

Logistics example1 2

P T

1

2

Truck

Drive(1,2) Drive(2,1)

Load(P,T,1)Unload(P,T,1)

Load(P,T, 1)Unload(P,T, 1)

1

2

T

Package

Load(P,T, 1)

Load(P,T, 2)

unload(P,T, 1)

unload(P,T, 2)

States are described by state variables

16

Logistics example1 2

1

2

Truck

Drive(1,2) Drive(2,1)

Load(P,T,1)Unload(P,T,1)

Load(P,T, 1)Unload(P,T, 1)

1

2

T

Package

Load(P,T, 1)

Load(P,T, 2)

unload(P,T, 1)

unload(P,T, 2)

Actions are state transformation functions

Effect

Prevail

17

One state change (1SC)

• Network representation

• Logistics example

2

1

2

1

2

1

2

1

t t

Truck

Package

h

g

f

h

g

f

h

g

f

t = 1

Prevail

Effect

Planning involves considering plans of increasing length

Plan step

18

One state change (1SC)

• Network representation

• Logistics example

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

t t t t

Drive(1,2)

Load(P,T, 1) Unload(P,T, 2)-

Load(P,T, 1) Unload(P,T, 2)Truck

Package

h

g

f

h

g

f

h

g

f

t = 1 t = 2 t = 3

Prevail

Effect

19

1SC formulation

• Constraints

– State changes (network flow), for all c C

gC ycf,g,t = 1{f I} for f Dc

hC ycg,h,t+1 = fC yc

g,h,t for f Dc , 1

t < T

fC ycf,g,T = 1 for g G

– Effect implications, for all c C, 1 t T

aA:(f,g)SC(a) xa,t = ycf,g,t for f, g Dc, f g

xa,t ycf,f,t for 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

0.01

0.1

1

10

100

1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Zenotravel

Tim

e (s

eco

nd

s)

SAT4

1SC0.01

0.1

1

10

100

1000

10000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1920 21222324 25262728

Logistics

So

luti

on

tim

e (s

ec.)

SAT4

1SC

0.01

0.1

1

10

100

1000

10000

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435

Blocksmove

So

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ec.)

SAT4

1SC

0.01

0.1

1

10

100

1000

1 2 3 4 5 6 7 8 9 1011121314151617181920

Driverlog

So

luti

on

tim

e (s

ec.)

SAT4

1SC

22

2. IP versus SAT approaches

1. 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

2. 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

2

1

2

1

2

1

2

1

2

1

2

1

t t t

Load(P,T, 1)Drive(1,2)

Load(P,T, 1) Unload(P,T, 2)

Unload(P,T, 2)

h

g

h

ff

g

Truck

Package

h

g

f

t = 1 t = 2

Prevail

Effect

24

Implied precedences (G1SC)

• Example

A1,A2A3

A4A1

A2

A3 A4

Implied precendence graph

25

Implied precedences (G1SC)

• Example

• Ordering (cycle elimination) constraints ensure a feasible ordering of the actions

A1,A2A3

A4

A4

A1

A1

A2

A3 A4

xA1,t + xA3,t + xA4,t 2

Implied precendence graph

26

G1SC formulation

• Constraints– State changes (network flow), for all c C

gC ycf,g,t = 1{f I} for f Dc

hC ycg,h,t+1 = fC yc

g,h,t for f Dc, 1 t T

fC ycf,g,T = 1 for g G

– Effect implications, for all c C, 1 t TaA:(f,f)SC(a) xa,t= yc

f,g,t for f, g Dc, f g,

xa,t ycf,f,t + gDc:f≠g (yc

g,f,t + ycf,g,t) for a A, f PR(a)

– Ordering (Cycle elimination) constraints

aV() xa,t |V()| – 1 for all cycles G, 1 t T

27

Branch-and-cut

Initialize LP

Node selection

LP solver

Branching

Cut generation

START

Nodes found?

Cuts found?

STOP

Feasible? Fathom

Z_lp < Z*?

no

yes

no

no

yes

yes

yes

no

Integer?

no

yes

28

Prevail

Effect

State change path (PathSC)

• Network representation

• Example

2

1

2

1

2

1

2

1

t t

Load(P,T, 1)Drive(1,2)

Unload(P,T, 2)

load(P,T, 1)unload(P,T,2)

h

g

f

h

g

Truck

Package

h

g

f f

t = 1

29

Summary of results

0

4

8

12

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Driverlog

Pla

n le

ng

th

SAT4

1SC

G1SC

PathSC

0

3

6

9

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Zenotravel

Pla

n le

ng

th

SAT4

1SC

G1SC

PathSC

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Logistics

Pla

n le

ng

th

SAT4

1SC

G1SC

PathSC

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11121314 1516 17181920212223242526272829303132333435

Blocksmove

Pla

n l

eng

th

SAT4

1SC

G1SC

PathSC

30

Summary of results

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1000

10000

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Logistics

So

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ec.)

SAT4

1SC

G1SC

PathSC

[van den Briel, Vossen, and Kambhampati, 2005, 2008]

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SAT4

1SC

G1SC

PathSC

0.01

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10

100

1000

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Problems (Zenotravel)

Tim

e (s

eco

nd

s)

SAT4

1SC

g1SC

kSC

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10

100

1000

10000

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Driverlog

So

luti

on

tim

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ec.)

SAT4

1SC

G1SC

PathSC

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]

PLAN EXISTENCE

PLAN LENGTH

PSP GOAL LENGTH

PSP GOAL

PLAN COST PSP UTILITY

PSP NET BENEFIT

Total Satisfaction

Problems

Partial Satisfaction

Problems

PSP UTILITY COST

33

Framework for optimal planning

• For step-based IP formulations optimality is restricted to the length of the plan

Plan step

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

t t t t

Drive(1,2)

Load(P,T, 1) Unload(P,T, 2)-

Load(P,T, 1) Unload(P,T, 2)Truck

Package

t = 1 t = 2 t = 3

34

Framework for optimal planning

1 2

P T

1

2

Truck

Drive(1,2) Drive(2,1)

Load(P,T,1)Unload(P,T,1)

Load(P,T, 1)Unload(P,T, 1)

1

2

T

Package

Load(P,T, 1)

Load(P,T, 2)

unload(P,T, 1)

unload(P,T, 2)

35

Action selection formulation

• Variables– xa Z+, for a A; xa is equal to the number of times action a is

executed

– yv(c,a) Z+, for v V, a A, a –(c); yv(c,a) is equal to the number of times transition v(c,a) is executed

• Objective function– MIN aA caxa

• Constraints av+(e) yv(c,a) – a v–(e) yv(c,a)

av+(e) yv(c,a) = xa

1 if c c0,v, c g–1 if c = c0,v, c g0 otherwise

No time indicesNo upper bounds

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 := (D1D2, A1A2, 1||2, 1||2, (c0,1, c0,2), g1g2)

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)\A1 undefined otherwise

38

Logistics example

1 2

P T

1

2

Truck

Drive(1,2) Drive(2,1)

Load(P,T,1)Unload(P,T,1)

Load(P,T, 1)Unload(P,T, 1)

1

2

T

Package

Load(P,T, 1)

Load(P,T, 2)

unload(P,T, 1)

unload(P,T, 2)

39

Simple logistics example

1,1

1,T

2,T

2,2

1,2

2,1

Truck || Package

Drive(1,2)

Drive(2, 1)

Load(P, T, 1)

Load(P, T, 2)

Unload(P, T, 1)

Unload(P, T, 2)

Drive(1, 2)

Drive(2, 1)

Drive(1, 2)Drive(2, 1)

1 2

P T

40

Summary of results

Highlighted values equal optimal solution

Problem h+ LP LP+ Optimalzenotravel1 1 1 1 1zenotravel2 4 3.0* 6 6zenotravel3 5 4.0* 6 6zenotravel4 6 5.0* 8 8zenotravel5 11 8.0* 11 11zenotravel6 11 8.0* 11 11zenotravel13 23 18.0* 24 -zenotravel19 62 46.0* 66.2* -zenotravel20 - 50.0* 68.3* -tpp1 4 3.0* 5 5tpp2 7 6.0* 8 8tpp3 10 9.0* 11 11tpp4 13 12.0* 14 14tpp5 17 15.0* 19 19tpp6 21 21.0* 25 25tpp28 - 150.0* - -tpp29 - - - -tpp30 - 174.0* - -bw-sussman 5 4 4 6bw-12step 4 4 4 12bw-large-a 12 12 12 12bw-large-b 16 16 16 18

Problem h+ LP LP+ Optimallog4-0 19 16.0* 20 20log4-1 17 14.0* 19 19log4-2 13 10.0* 15 15log5-1 15 12.0* 17 17log5-2 8 6.0* 8 8log6-1 13 10.0* 14 14log6-9 21 18.0* 24 24log12-0 39 32.0* 42 42log15-1 63 54.0* 67 -freecell2-1 9 9 9 9freecell2-2 8 8 8 8freecell2-3 8 8 8 8freecell2-4 8 8 8 8freecell2-5 9 9 9 9freecell3-5 13 12 12 -freecell13-3 - 55 55 -freecell13-4 - 54 54 -freecell13-5 - 52 52 -driverlog1 6 3.0* 7 7driverlog2 14 12.0* 19 19driverlog3 11 8.0* 11 12driverlog4 12 11.0* 16 16driverlog6 10 8.0* 11 11driverlog7 12 11.0* 13 13driverlog13 21 15.0* 24 -driverlog19 89 60.0* 96.6* -driverlog20 84 60.0* 89.5* -

41

Summary of results

0.01

0.1

1

10

100

1000h+

LP

LP+

Logistics Freecell Driverlog Zenotravel TPP Blocks

42

Utilize LP in heuristic search

0

100000

200000

300000

400000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Zenotravel

Net

ben

efit

H_LP

H_LP + RP

H_LP + Cost RP

UB

0

300000

600000

900000

1200000

1 3 5 7 9 11 13 15 17 19

Satellite

Net

ben

efit

H_LP

H_LP + RP

H_LP + Cost RP

UB

0

100000

200000

300000

400000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Rovers

Net

ben

efit

H_LP

H_LP + RP

H_LP + Cost RP

UB

[Benton, van den Briel, and Kambhampati, 2007]

BBOP-LP planner

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 31

Cited by 15

Cited by 3

Cited by 4

Cited by 6

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

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