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Reviving Integer Programming Approaches for AI Planning: A Branch-and-Cut Framework
Thomas VossenLeeds School of Business
University of Colorado at BoulderBoulder, CO
Subbarao KambhampatiDepartment of Computer Science
Arizona State UniversityTempe, AZ
Menkes van den BrielDepartment of Industrial
EngineeringArizona State University
Tempe, [email protected]
Compilation approaches for AI planning
Integer Programming
(IP)
PropositionalSatisfiability
(SAT) ConstraintSatisfaction
(CSP)
Planning
• Very little focus on IP
• There were no IP-based planners in IPCs 1998, 2000, 2002
Background
• Why should we pursue integer programming?– Support for optimality– Support for numeric constraints– IP has shown to be a great success in solving
large scale optimization problems
Reviving integer programming
1. State variables are modeled by an appropriately defined network– The resulting IP formulation can be interpreted as a
network flow problem with additional side constraints
2. The common notion of parallelism based on planning graphs is generalized– This reduces the plan length of the solution plan
3. Action ordering constraints are dynamically generated when needed by a branch-and-cut algorithm– Branch-and-cut has been extremely successful for a
large number of optimization problems
Example: State change flow network
LOC1 LOC2
Example: State change flow network
I
I
G
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
t = 1 t = 2 t = 3
LOC1 LOC2
0 1 2 3
Package1
Truck1
Example: State change flow network
I
I
G
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
t = 1 t = 2 t = 3
LOC1 LOC2
0 1 2 3
Package1
Truck1
Action effects
• STRIPS (binary-valued)
• SAS+ (multi-valued)
LOAD(Package1,Loc1)
PRE: Package1_AT_LOC1, Truck1_AT_LOC1
DEL: Package1_AT_LOC1
ADD: Package1_IN_TRUCK1
LOAD(Package1,Loc1)
State change effects Package1: AT_LOC1 IN_TRUCK1
Prevail conditions Truck1: AT_LOC1 AT_LOC1
LOC1
Prevail condition
State change effect
Example: state change flow network
I
I
G
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
t = 1 t = 2 t = 3
LOC1 LOC2
Action effects link multiple networks together
Package1
Truck1
LOAD(Package1) DRIVE(Truck1,Loc1,Loc2) UNLOAD(Package1)
Single state change (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)SCa xa,t = ycf,g,t for f, g Dc, f g
xa,t ycf,f,t for aA, (f,f)PRa
Reviving integer programming
1. State variables are modeled by an appropriately defined network– The resulting IP formulation can be interpreted as a
network flow problem with additional side constraints
2. The common notion of parallelism based on planning graphs is generalized– This reduces the plan length of the solution plan
3. Action ordering constraints are dynamically generated when needed by a branch-and-cut algorithm– Branch-and-cut has been extremely successful for a
large number of optimization problems
Generalized parallelism
• Common notion of parallelism (Graphplan)– Actions can be executed in the same plan step as long
as they do not interfere with each other (no action deletes the precondition or add effect of another action)
– In an actual plan it is possible to arrange independent parallel actions in any order with exactly the same outcome
• Alternative concept of parallelism– Allow actions to be executed in the same plan step as
long as there exists a feasible ordering of the actions such that: • All the actions preconditions are met• Within each state variable and for each plan step,
the value is changed at most once
Generalized parallelism
Increase in parallelism requires “generalized” state
changes
Graphplan parallelism Alternative parallelism
Example: Generalized parallelism
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
Package1
Truck1
Action effect implications
LOAD(Package1,Truck1) Increase in
parallelism requires “generalized” state
changes
LOAD(Package1,Truck1)
DRIVE(Truck1,Loc1,Loc2)
Implied precedences
• Implied precedences
A1,A2A3
A4
A1
A2
A3 A4
Implied precedence graph
Implied precedences
• Implied precedences
• Cycle elimination constraints need to ensure feasible orderings
A1,A2A3
A4
A4
A1
A1
A2
A3 A4
Implied precedence graph
Generalized single state change (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 yc
f,g,T = 1 for g G Dc
– Effect implications, for all c C, 1 t TaA:(f,f)SCa 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 aA, (f,f)PRa
– Ordering (Cycle elimination) constraints
aV() xa,t |V()| – 1 for all cycles G, 1 t T
Reviving integer programming
1. State variables are modeled by an appropriately defined network– The resulting IP formulation can be interpreted as a
network flow problem with additional side constraints
2. The common notion of parallelism based on planning graphs is generalized– This reduces the plan length of the solution plan
3. Action ordering constraints are dynamically generated when needed by a branch-and-cut algorithm– Branch-and-cut has been extremely successful for a
large number of optimization problems
Branch-and-cut algorithm
• Branch-and-cut (row generation with branch-and-bound)– Classes of valid inequalities are left out of the LP
relaxation because there are too many constraints to handle efficiently, and most of them will not be binding in an optimal solution anyway
– Branching occurs when no violated inequalities in a class are found and the LP solution does not satisfy the integrality constraints
Reviving integer programming
1. State variables are modeled by an appropriately defined network– The resulting IP formulation can be interpreted as a
network flow problem with additional side constraints
2. The common notion of parallelism based on planning graphs is generalized– This reduces the plan length of the solution plan
3. Action ordering constraints are dynamically generated when needed by a branch-and-cut algorithm– Branch-and-cut has been extremely successful for a
large number of optimization problems
Further generalizations
• Allow actions to be executed in the same plan step as long as there exists a feasible ordering of the actions such that – All the actions preconditions are met– For each state variable, more than one state
change is allowed, e.g. “state change paths”
I
I
G
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
t = 1 t = 2
0 0’ 1 1’
Package1
Truck1
Further generalizations
• There exists a feasible ordering of the actions such that – All the actions preconditions are met– For each state variable, more than one state
change is allowed, e.g. “state change paths”
I
G
I
AT_LOC1
AT_LOC2
IN_TRUCK1
AT_LOC1
AT_LOC2
t = 1
0 0’
Package1
Truck1
LOAD(Package1) DRIVE(Truck1,Loc1,Loc2) UNLOAD(Package1)
State change path (kSC) formulation
• Constraints– State changes (network flow), for all c C
gC,fg ycg,f,1 + 1 {f I} = yc
f,f,1 yc
f,f,1 = gC,fg ycf,g,1 + zc
f,1 for f CgC,fg yc
g,f,1 + zcf,t-1 = yc
f,f,t yc
f,f,t = gC,fg ycf,g,t + zc
f,t for f C, 1 t T – 1 gC,fg yc
g,f,T + zcf,T-1 = yc
f,f,T yc
f,f,T = gC,fg ycf,g,T + zc
f,T for f C
– Effect implications, for all c C, 1 t T
aA:(f,f)SCa xa,t= ycf,g,t for f, g Dc, f g,
xa,t ycf,f,t + gDc:f≠g (yc
g,f,t + ycf,g,t) for aA, (f,f)PRa
– Ordering (Cycle elimination) constraints aV() xa,t |V()| – 1 for all cycles G, 1 t T
Experimental Setup
• Four domains from the International Planning Competition – Blocks-world, Logistics, Driverlog, Zenotravel– Used Fast Downward preprocessor (translates PDDL to multi-
valued state variables)
• Compared results of 1SC, G1SC, and kSC to SATPLAN04 (SAT4) – State-of-the art SAT-based planner– First prize in the IPC-2004
• Resources– 2.67GHz Linux machine, 1Gb memory– ILOG CPLEX 8.1– 30 minute maximum for each problem
Performance Quality
0.01
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 19 20 21 22 23 24 25 26 27 28
Problems (Logistics)
Tim
e (s
eco
nd
s)
SAT4
1SC
G1SC
kSC
0.01
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 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Problems (Blocksmove)
Tim
e (s
eco
nd
s)
SAT4
1SC
G1SC
kSC
0
5
10
15
20
25
30
35
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 29 30 31 32 33 34 35
Problems (Blocksmove)
Pla
n le
ng
th (
# o
f ac
tio
ns)
SAT4
1SC
G1SC
kSC
0
20
40
60
80
100
120
140
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
Problems (Logistics)
Pla
n le
ng
th (
# o
f ac
tio
ns)
SAT4
1SC
G1SC
kSC
Performance Quality
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Problems (Driverlog)
Pla
n le
ng
th (
# o
f ac
tio
ns)
SAT4
1SC
g1SC
kSC
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
Problems (Zenotravel)
Tim
e (s
eco
nd
s)
SAT4
1SC
G1SC
kSC
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Problems (Zenotravel)
Pla
n le
ng
th (
# o
f ac
tio
ns)
SAT4
1SC
G1SC
kSC
0.01
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 19 20
Problems (Driverlog)
Tim
e (s
eco
nd
s)
SAT4
1SC
g1SC
kSC
0
2
4
6
8
10
12
14
16
18
Pla
n st
eps
SAT4
1SC
G1SC
kSC
Blocks Logistics Driverlog Zenotravel
Results: plan length
Summary and conclusions
• First integer programming formulation that is competitive with state-of-the art SAT-based planner:– Uses multi valued states and represent state changes as
flows in an appropriately defined network– Uses generalized notion of parallelism in a branch-and-cut
algorithm
• Future work– Optimal planning (minimum number of actions, minimum
cost plans, minimum resource usage, minimum makespan)
– Expand the scope of planning problems, apply integer programming techniques to planning with resources and temporal planning
