Introduction
Problem Description
Results
Planning Complexity—A Parameterized
Analysis
Yue Chen
Vienna University of Technology
Joint work with Christer Bäckström, Peter Jonsson,
Sebastian Ordyniak and Stefan Szeider
February 2012
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
A little history to begin with
Standard Complexity Analysis
The computational complexity of propositional STRIPS planning
by Tom Bylander, 1994
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
A little history to begin with
Standard Complexity Analysis
The computational complexity of propositional STRIPS planning
by Tom Bylander, 1994
It is PSPACE-complete to determine if any given
planning instance has any solutions.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Parameterized complexity analysis
Standard Complexity Analysis
I
Bylander
I
Bäckström and Nebel
I
P, U, B, S
I
...
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Parameterized complexity analysis
Standard Complexity Analysis
Parameterized
Complexity Analysis
I
Bylander
I
Bäckström and Nebel
I
Stefan Szeider
I
P, U, B, S
I
I
...
Sebastian
Ordyniak
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
An example of planning
Tower of Hanoi
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
The language of planning
Definition
An instance of BOUNDED SAS + PLANNING problem P is a
tuple P = (V , D, A, I, G) with components defined as follows:
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
The language of planning
Definition
An instance of BOUNDED SAS + PLANNING problem P is a
tuple P = (V , D, A, I, G) with components defined as follows:
I
V = {v1 , . . . , vn } is a set of state variables. D is a domain
function for V . Each variable has an associated domain
which implicitly defines an extended domain
Dv+ = Dv ∪ {u}, where u denotes the undefined value. A
state s is an n-tuple of values. The states that contain u
are called partial states, otherwise total states. s[v ]
denotes the value of the variable v in a state s.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
The language of planning Cont’d
Definition
I
A is a set of actions. Each action a ∈ A has a precondition
pre(a) and an effect eff(a), both are partial states.
I
I as a total state is the initial state and G as a partial state
is the goal state.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Preliminary definitions
Let a be an action and
s is a state,
I
a is valid in s if
pre(a) is either s
or undefined w.r.t
all variables;
I
The result of a in s
is the state s
updated by the
effect of a;
I
A state s is a goal
state iff s equals G
unless G is
undefined w.r.t all
variables.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Preliminary definitions
Let a be an action and
s is a state,
I
a is valid in s if
pre(a) is either s
or undefined w.r.t
all variables;
I
The result of a in s
is the state s
updated by the
effect of a;
I
A state s is a goal
state iff s equals G
unless G is
undefined w.r.t all
variables.
Let s0 , sl be two total states,
ω = ha1 , . . . , an i is a possibly
empty sequence of actions. ω is a
plan from s0 to sl iff
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Preliminary definitions
Let a be an action and
s is a state,
I
a is valid in s if
pre(a) is either s
or undefined w.r.t
all variables;
I
The result of a in s
is the state s
updated by the
effect of a;
I
A state s is a goal
state iff s equals G
unless G is
undefined w.r.t all
variables.
Let s0 , sl be two total states,
ω = ha1 , . . . , an i is a possibly
empty sequence of actions. ω is a
plan from s0 to sl iff
I
either ω is an empty sequence
or
I
there are l − 1 total states
such that there is a valid
action in each of them, and the
result of that action updates
the states in which it is valid to
its successor in the sequence.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Preliminary definitions
Let a be an action and
s is a state,
I
a is valid in s if
pre(a) is either s
or undefined w.r.t
all variables;
I
The result of a in s
is the state s
updated by the
effect of a;
I
A state s is a goal
state iff s equals G
unless G is
undefined w.r.t all
variables.
Let s0 , sl be two total states,
ω = ha1 , . . . , an i is a possibly
empty sequence of actions. ω is a
plan from s0 to sl iff
I
either ω is an empty sequence
or
I
there are l − 1 total states
such that there is a valid
action in each of them, and the
result of that action updates
the states in which it is valid to
its successor in the sequence.
ω is a plan for P iff it is a plan from I
to some goal state.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
The SAS + BOUNDED PLANNING Problem
Instance: A pair hP, k i where P is an SAS + instance and k a
positive integer.
Parameter: k
Question: Is there a plan for P of length at most k ?
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Syntactic Restrictions
A BOUNDED SAS + PLANNING instance is:
(P) post-unique if no two distinct actions can change the same
state variable to the same value;
(U) unary if each action changes exactly one state variable;
(B) binary if |D| = 2;
(S) single-valued if any two actions that both change the same
state variable from defined to undefined value must require
the same defined value.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Hardness Results for Bounded SAS + Planning with
Syntactic Restrictions
I
{B, S}-BOUNDED SAS + PLANNING is W[2]-hard when
the actions have no preconditions;
I
{U, B, S}-BOUNDED SAS + PLANNING is W[1]-hard when
every actions has at most one precondition and one effect;
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Membership Results for Bounded SAS + Planning with
Syntactic Restrictions
I
BOUNDED SAS + PLANNING is in W[2];
I
{U}-BOUNDED SAS + PLANNING is in W[1];
I
{P}-BOUNDED SAS + PLANNING is in FPT;
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Completeness Results for Bounded SAS + Planning
I
{B, S}-BOUNDED SAS + PLANNING is W[2]-complete
when the actions have no preconditions;
I
{U, B, S}-BOUNDED SAS + PLANNING is W[1]-complete
when every actions has at most one precondition and one
effect;
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
PUBS
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PSPACE-C
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
in P
NP-C
PUBS
NP-H
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
in FPT
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
W[1]-C
PUBS
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PSPACE-C
in FPT
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
in P
W[1]-C
NP-C
PUBS
NP-H
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
in P
PUBS
NP-H
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
in FPT
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
in P
NP-C
PUBS
NP-H
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
W[1]-C
PUBS
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
W[1]-C
NP-C
PUBS
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
PSPACE-C
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
W[1]-C
PUBS
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
PUBS Lattice with Old and New Result
-
in FPT
PU
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
BS
W[2]-C
in P
W[1]-C
NP-C
PUBS
NP-H
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
Thank You!
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
s[v ] = 1
a
s[v ] = 0
a0
s[v ] = u
s
..
.
s0
..
.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
v1
..
.
vn
a
v1
..
.
s
a
s0
vn
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider
Introduction
Problem Description
Results
s[v1 ] = 1
a
s[v1 ] = 0
s
a0
..
.
s0
u
..
.
Yue Chen Vienna University of Technology Joint work with Christer Bäckström,
Planning Complexity—A
Peter Jonsson,Parameterized
Sebastian Ordyniak
Analysis
and Stefan Szeider