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An LPBased Heuristic for Optimal Planning

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Title: An LPBased Heuristic for Optimal Planning


1
An LP-Based Heuristic for Optimal Planning
Menkes van den BrielDepartment of Industrial
EngineeringArizona State Universitymenkes_at_asu.ed
u
J. BentonDepartment of Computer ScienceArizona
State Universitybentonj_at_asu.edu
Subbarao KambhampatiDepartment of Computer
ScienceArizona State Universityrao_at_asu.edu
Thomas VossenLeeds School of BusinessUniversity
of Colorado at Bouldervossen_at_colorado.edu
http//rakaposhi.eas.asu.edu/yochan/
2
What is automated planning?
loc1
loc2
Initial states0 ? S
Goals ? S
3
What is automated planning?
loc1
loc2
Initial states0 ? S
Goals ? S
loc1
loc1
Action
a ?pre, post, prevail?
4
What is automated planning?
loc1
loc2
Initial states0 ? S
Goals ? S




PlanP ?a1, , an?
loc1
loc1
Action
a ?pre, post, prevail?
5
Motivation
  • Why heuristics?
  • Heuristic state space search have been very
    successful in solving automated planning problems
  • Why optimal planning?
  • Real-world planning applications require optimal
    or near-optimal solutions
  • The difference between a (near) optimal solution
    and a feasible solution may be the difference
    between winning or losing the interest of an
    investor or strategic partner

6
LP-based heuristic
Relax the ordering of the actions
Setup an integer programming formulation
Solve the LP-relaxation and use the objective
function value as an admissible distance estimate
Strengthen the formulation by adding valid
inequalites
7
Action selection formulation
  • Represent the planning problem as a set of
    loosely coupled network flow problems
  • Each state variable defines one network flow
    problem
  • Nodes correspond to the state variable values
  • Arcs correspond to state variable transitions

8
Simple logistics example
loc1
loc2
DTGTruck1
Load(p1,t1,l1)Unload(p1,t1,l1)
1
Drive(l1,l2)
Drive(l2,l1)
2
Load(p1,t1,l1)Unload(p1,t1,l1)
DTGPackage1
1
Load(p1,t1,l1)
Unload(p1,t1,l1)
2
Load(p1,t1,l2)
Unload(p1,t1,l2)
T
9
Action selection formulation
  • Variables
  • xa ? Z, for a ? A xa is equal to the number of
    times action a is executed
  • Objective function
  • MIN ?a?A xa
  • Constraints, for all c ? C, f ? Vc
  • ?e?Vc(f)a?AcE(e) xa ?e?Vc(f)b?AcE(e) xb ?
  • xa ? M ?e?Vc(f)b?AcE(e) xb for all f ? s0c, a
    ? AcV(f)

No time indicesNo upper bound
1 if f ? s0c, f sc1 if f s0c, f ?
sc0 otherwise
10
Simple logistics example
loc1
loc2
DTGTruck1
Load(p1,t1,l1)Unload(p1,t1,l1)
1
Drive(l1,l2)
Drive(l2,l1)
2
Load(p1,t1,l1)Unload(p1,t1,l1)
DTGPackage1
1
Load(p1,t1,l1)
Unload(p1,t1,l1)
2
Load(p1,t1,l2)
Unload(p1,t1,l2)
T
11
Simple logistics example
Drive(l2,l1)
Load(p1,t1,l1)
Drive(l1,l2)
Unload(p1,t1,l2)
DTGTruck1
Load(p1,t1,l1)Unload(p1,t1,l1)
1
Feasible plan xDrive(l2,l1) 1xLoad(p1,t1,l1)
1xDrive(l1,l2) 1xUnload(p1,t1,l2) 1
Drive(l1,l2)
Drive(l2,l1)
2
Load(p1,t1,l1)Unload(p1,t1,l1)
DTGPackage1
1
Load(p1,t1,l1)
Unload(p1,t1,l1)
4
2
Load(p1,t1,l2)
Unload(p1,t1,l2)
T
12
Simple logistics example


Drive(l2,l1)
Load(p1,t1,l1)
Unload(p1,t1,l2)
DTGTruck1
Load(p1,t1,l1)Unload(p1,t1,l1)
1
LP solution xLoad(p1,t1,l1) 1xUnload(p1,t1,l2)
1xDrive(l2,l1) 1/M
Drive(l1,l2)
Drive(l2,l1)
2
Load(p1,t1,l1)Unload(p1,t1,l1)
DTGPackage1
1
Load(p1,t1,l1)
Unload(p1,t1,l1)
2 1/M
2
Load(p1,t1,l2)
Unload(p1,t1,l2)
T
13
Preliminary results
14
Preliminary results
15
Strengthening techniques
  • Composition of state variables (i.e. fluent
    merging)
  • Given the domain transition graph (DTG) of two
    state variables c1, c2, the composition of DTGc1
    and DTGc2 is the domain transition graph
    DTGc1c2 (Vc1c2, Ec1c2) where
  • Vc1c2 Vc1 ? Vc2
  • ((f1,g1),(f2,g2)) ? Ec1c2 if f1,f2 ? Vc1, g1,g2
    ? Vc2 and there exists an action a ? A such that
    one of the following conditions hold
  • prec1 f1, postc1 f2, and prec2 g1,
    postc2 g2
  • prec1 f1, postc1 f2, and prevailc2
    g1, g1 g2
  • prec1 f1, postc1 f2, and g1 g2

The term composition is also used in model
checking to define the parallel composition or
the synchronized product of automata Cassandras
Lafortune, 1999
16
Example
  • Two DTGs and their composition

f1,,g1
d
f1,g2
f3,g2
c
f1
a
a
c
c
f2,g1
f3,g1
f2
g1
a
b
b
d
b
d
f2,g2
f3
g2
DTGc1
DTGc2
DTGc1 c2
17
Example
  • Two DTGs and their composition
  • Small in-arcs denote the initial state
  • Double circles denote the goal

f1,,g1
d
f1,g2
c
f1
a
a
c
c
f2,g1
f3,g1
f2
g1
a
b
b
d
b
d
f2,g2
f3
g2
DTGc1
DTGc2
DTGc1 c2
18
Simple logistics example
loc1
loc2
DTGTruck1 Package1
2,1
Drive(l2,l1)
Drive(l1,l2)
1,1
1,2
Load(p1,t1,l1)
Unload(p1,t1,l1)
Drive(l1,l2)
Drive(l2,l1)
1,T
2,2
Unload(p1,t1,l2)
Drive(l2,l1)
Load(p1,t1,l2)
Drive(l1,l2)
2,T
19
Simple logistics example
Drive(l2,l1)
Load(p1,t1,l1)
Drive(l1,l2)
Unload(p1,t1,l2)
DTGTruck1 Package1
2,1
LP solution xDrive(l2,l1) 1xLoad(p1,t1,l1)
1xDrive(l1,l2) 1xUnload(p1,t1,l2) 1
Drive(l2,l1)
Drive(l1,l2)
1,1
1,2
Unload(p1,t1,l1)
Drive(l1,l2)
Drive(l2,l1)
1,T
2,2
Unload(p1,t1,l2)
Drive(l2,l1)
4
Load(p1,t1,l2)
Drive(l1,l2)
2,T
20
Another example
  • Two DTGs and their composition

f1,,g1
f1,g2
f3,g3
f1
g1
f1,g3
f3,g2
f2
g2
f2,g1
f3,g1
f2,g2
f2,g3
f3
g3
DTGc1
DTGc2
DTGc1 c2
21
Another example
  • Two DTGs and their composition
  • Solution to the individual state variables

f1,,g1
f1,g2
f3,g3
f1
g1
f1,g3
f3,g2
a
b
f2
g2
f2,g1
f3,g1
b
a
f2,g2
f2,g3
f3
g3
DTGc1
DTGc2
DTGc1 c2
22
Another example
  • Two DTGs and their composition
  • Solution to the individual state variables
    represented in the composed state variable

f1,,g1
f1,g2
f3,g3
a
f1
g1
f1,g3
f3,g2
b
a
b
f2
g2
f2,g1
f3,g1
b
a
f2,g2
f2,g3
f3
g3
DTGc1
DTGc2
DTGc1 c2
23
Another example
  • Two DTGs and their composition
  • Solution to the individual state variables
    represented in the composed state variable

f1,,g1
f1,g2
f3,g3
a
f1
g1
f1,g3
f3,g2
b
a
b
f2
g2
f2,g1
f3,g1
b
a
f2,g2
f2,g3
f3
g3
DTGc1
DTGc2
DTGc1 c2
Violates balance of flow constraints
24
Another example
  • Two DTGs and their composition
  • Adding new balance of flow constraints
    strengthens the formulation

f1,,g1
c
f1,g2
f3,g3
a
e
f1
g1
f1,g3
f3,g2
b
a
b
c
f2
g2
d
f2,g1
f3,g1
d
b
a
e
f2,g2
f2,g3
f3
g3
DTGc1
DTGc2
DTGc1 c2
25
Identifying mergeable fluents
  • When should we create a composition of two or
    more state variables?
  • Look at the causal graph
  • Look at the actions that introduce dependencies
    in the causal graph

Person 1
Person 2
Person 1
Person 2
Airplane 1
Airplane 2
Airplane 1 Fuel1
Airplane 2 Fuel2
Fuel 1
Fuel 2
26
Experimental setup
  • Objective
  • Minimize number of actions
  • Domains
  • Selected domains from the International Planning
    Competition
  • Logistics
  • Freecell
  • Driverlog
  • Zenotravel
  • TPP
  • Blocksworld
  • Resources
  • 2.67Ghz Linux machine
  • 1GB memory
  • 15 minutes runtime
  • CPLEX 10.0

27
Experimental setup
  • Distance estimates
  • LP
  • Action selection formulation with strengthening
  • LP
  • Action selection formulation without
    strengthening
  • Lplan
  • Step based integer programming formulation by
    Lplan Bylander, 1997
  • h
  • Optimal relaxed plan when the delete effects are
    ignored
  • hFF
  • Inadmissible but efficient relaxed plan heuristic
    by FF Hoffmann, and Nebel, 2001
  • Optimal
  • Optimal distance estimate given by Satplanner
    using the opt flag Rintanen, Heljanko, and
    Niemela, 2005

28
Experimental results
29
Experimental results
Distance estimates from the initial state to the
goal (highlighted values equal the optimal
distance)
30
Experimental results
  • Heuristic calculation time

31
Conclusions and future work
  • LP-based heuristic that respects delete effects,
    but ignores action ordering shows very promising
    results
  • Finds the optimal distance estimate in several
    problem instances
  • Can be used to calculate admissible distance
    estimates for various optimization problems in
    planning
  • Ongoing work successfully incorporated our
    LP-based heuristic in a search algorithm that
    solves oversubscription planning
  • Interesting directions for future work
  • Apply fluent merging more aggressively
  • Extend the formulation into a complete planning
    system

32
LP-based heuristic
Relax the ordering of the actions
Setup an integer programming formulation
Solve the LP-relaxation and use the objective
function value as an admissible distance estimate
Strengthen the formulation by adding valid
inequalites
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