“poster” — 2012/7/18 — 14:14 — page 1 — #1
AAAI 2012, Toronto, Canada
The Complexity of Planning Revisited
A Parameterized Analysis
Christer Bäckström1, Yue Chen2, Peter Jonsson1, Sebastian Ordyniak2 and Stefan Szeider2
B OUNDED SAS+ P LANNING
Our Approach
+
Problem: Given an SAS instance P and a positive integer l find a plan of
length at most l (if such a plan exists).
Goal: Obtain a more fine grained Complexity Landscape for B OUNDED
+
SAS P LANNING by studying the parameterized complexity of B OUNDED
SAS+ P LANNING with respect to the length of the plan.
Syntactical Restrictions
Method: We revisit B OUNDED SAS+ P LANNING in the context of
Parameterized Complexity Theory. Using the length of a plan as a
parameter, we obtain a detailed map of the parameterized complexity of
B OUNDED SAS+ P LANNING. We complement the analysis by applying
the same syntactical restrictions as in the classical setting.
To identify tractable sub problems for B OUNDED SAS+ P LANNING several
syntactical restrictions have been introduced by Backstrom and Klein (CI
+
1991). A B OUNDED SAS P LANNING instance is:
(P) POST-UNIQUE if no two actions change the same variable to the
same value;
(U) UNARY if each action changes exactly one variable;
(B) BINARY if every variable has exactly 2 values;
(S) SINGLE-VALUED if every two actions that depend on the same
variable (but do not change that variable) require the same value
from that variable.
Our Results
Combined Classical and Parameterized Complexity Landscape with
respect to the restrictions P, U,B, S:
P SPACE-C
Additionally, restrictions on the number of preconditions and effects have
been studied by Bylander (AI 1994).
W[2]-C
Parameterized Complexity Theory
P
U
S
B
PS
PB
US
UB
PUS
PUB
PBS
UBS
Aim: Parameterized complexity theory provides a more fine-grained
analysis of the complexity of problems than classical complexity theory.
Method: Whereas in classical complexity the complexity of a problem is
merely measured relative to its input size, parameterized complexity
takes into account any number of additional parameters. This allows to
identify the crucial parts of a problem, i.e., the parts that make it
intractable.
PU
NP-H
in P
Parameterized Complexity Classes:
FPT: A parameterized problem is
fixed-parameter tractable (or FPT) if
it can be solved by an algorithm
whose running time depends
arbitrarily on the parameter but only
polynomially on the input size
(O(f (k) + p(n))).
FPT is considered to be the
parameterized analogue of P.
W[1]–W[P]: W[?]-complete problems
are unlikely to be fixed-parameter
tractable but there is a non-uniform
polynomial time algorithm for these
k
problems (O(n )).
The W[?] classes are considered to
be the parameterized analogue of
NP.
W[1]-C
PUBS
W[P]
W[2]
W[1]
FPT
References
I
C. Bäckström, I. Klein, Planning in polynomial time: the SAS-PUBS class (CI 1991).
I
T. Bylander, The computational complexity of propositional strips planning (AI 1994).
I
C. Bäckström, B. Nebel, Complexity results for SAS+ planning (CI 1995).
1
BS
in FPT
Combined Classical and Parameterized Complexity Landscape with
respect to restrictions on the number of preconditions (mp) and the
number of effects (me):
fix me > 1 arb. me
mp = 0 in P
in W[1]
W[2]-C
in P
NP-C
NP-C
mp = 1 W[1]-C
W[1]-C
W[2]-C
NP-H
NP-H
P SPACE-C
fix mp > 1 W[1]-C
W[1]-C
W[2]-C
NP-H
P SPACE P SPACE
arb. mp
W[1]-C
W[1]-C
W[2]-C
P SPACE-C P SPACE-C P SPACE-C
Linköping University.
2
Vienna University of Technology. Research supported by the European Research Council (COMPLEX REASON, 239962).
me = 1