Framework and Complexity Results for ... - Semantic Scholar
Framework and Complexity Results for Coordinating Non-Cooperative Planning Agents
J. Renze Steenhuisen September 20, 2006 1
Delft University of Technology Delft University of Technology
Overview •
•
•
Variant I: Pure Coordination Variant II: Coordinated Assignment Variant III: Complete Coordination
September 20, 2006
Delft University of Technology
2
A Depot Example
A1
A2
t1
t3 d
t4
t2
September 20, 2006
Delft University of Technology
t1
t2
A1
A2
t4
t3
3
A Fixed Task Instance t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 4
Coordination (A Problem) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 5
Pure Coordination Recognition Problem: "Are there still potential directed cycles?" P URE C OORDINATION R ECOGNITION INSTANCE: Given a fixed task instance. QUESTION: Is this instance coordinated?
September 20, 2006
Delft University of Technology
6
A Fixed Task Instance t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 7
Coordination (A Solution) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 8
Coordination (A Solution) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 9
Pure Coordination Problem: "How can we exclude all possible cycles with a minimal number of additional constraints?" P URE C OORDINATION INSTANCE: Given a fixed task instance and integer K ≥ 0. QUESTION: Can this instance be coordinated by adding at most K arcs?
September 20, 2006
Delft University of Technology
10
Complexity Analysis •
•
P URE C OORDINATION R ECOGNITION CO NP-complete P URE C OORDINATION Σp2 -complete
September 20, 2006
Delft University of Technology
11
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
12
Subclasses of Pure Coordination Limiting the number of tasks per agent PCRT (n)
PCT (n)
n=2
P
NP-complete
n=3
P
NP-complete
n=4
CO NP-complete
Σp2
n=5
CO NP-complete
Σp2
n=6
CO NP-complete
Σp2
n≥7
CO NP-complete
Σp2 -complete
September 20, 2006
Delft University of Technology
13
Subclasses of Pure Coordination Limiting the number of agents PCRA(n)
PCA(n)
n=2
P
NP
n≥3
P
NP-complete
September 20, 2006
Delft University of Technology
14
Proof: PCRA(2) ∈ P t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3
15
Proof: PCRA(3) ∈ P
A1
t1
t
t6
t3
t5
September 20, 2006
Delft University of Technology
A2
2
t4
A3 16
Proof: 3VC ∝ PCA(3) u
v
z
September 20, 2006
Delft University of Technology
w
y
x
17
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
18
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
19
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
20
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
21
Proof: 3VC ∝ PCA(3) u
v
z
September 20, 2006
Delft University of Technology
w
y
x
22
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
23
The Depot Example
A1
A2
t1
t3 d
t4
t2
September 20, 2006
Delft University of Technology
t1
t2
A1
A2
t4
t3
24
Task Hierarchy Example
bike
t1
t2
OR
AND
car
truck
t 21
•
Tasks: requirements, decomposition
•
Agents: capabilities
September 20, 2006
Delft University of Technology
t 22
25
Coordinated Assignment Recognition Problem: "Is there an assignment without potential directed cycles?" C OORDINATED A SSIGNMENT R ECOGNITION INSTANCE: Given a free task instance. QUESTION: Does there exist a partitioning that is coordinated?
September 20, 2006
Delft University of Technology
26
Coordinated Assignment Problem: "How can we introduce an assignment that is free of potential directed cycles by adding a minimal number of constraints?" C OORDINATED A SSIGNMENT INSTANCE: Given a free task instance and integer K ≥ 0. QUESTION: Does there exist a partitioning that can be coordinated by adding at most K arcs? September 20, 2006
Delft University of Technology
27
Complexity Analysis •
•
C OORDINATED A SSIGNMENT R ECOGNITION p Σ2 -complete C OORDINATED A SSIGNMENT Σp2 -complete
September 20, 2006
Delft University of Technology
28
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
29
Complete Coordination Recognition Problem: "Are all assignments free of potential directed cycles?" C OMPLETE C OORDINATED R ECOGNITION INSTANCE: Given a free task instance. QUESTION: Does it hold for all possible partitions that the resulting fixed task instance is coordinated? September 20, 2006
Delft University of Technology
30
Complete Coordination Problem: "How can we make all assignments free of potential directed cycles by adding a minimal number of constraints?" C OMPLETE C OORDINATION INSTANCE: Given a free task instance and integer K ≥ 0. QUESTION: Can all possible partitions be coordinated by adding at most K arcs? September 20, 2006
Delft University of Technology
31
Complexity Analysis •
•
C OMPLETE C OORDINATION R ECOGNITION p Π2 -complete C OMPLETE C OORDINATION Πp3 -complete
September 20, 2006
Delft University of Technology
32
Polynomial Hierarchy
P
Σ 2 −complete P
Σ2
NP
coNP
NPc
P
P
Π 3 −complete
coNPc
P
Π2 P
Π 2 −complete
September 20, 2006
Delft University of Technology
P
Π3
33
Future Work •
Subclasses of C OORDINATED A SSIGNMENT
•
Subclasses of C OMPLETE C OORDINATION
•
•
Other constraints: • Time windows • Resource constraints Dynamically adding/removing constraints/tasks (i.e., towards coordination during planning)
September 20, 2006
Delft University of Technology
34
Framework and Complexity Results for Coordinating Non-Cooperative Planning Agents
J. Renze Steenhuisen September 20, 2006 35
Delft University of Technology Delft University of Technology
Proof: PC T (2 ) is NP-complete u
v
x
w
y
September 20, 2006
Delft University of Technology
z
36
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
37
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
38
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
39
Proof: PC T (2 ) is NP-complete u
v
x
w
y
September 20, 2006
Delft University of Technology
z
40
Proof: PCR T (4 ) is
CO NP-complete t
s u
x
September 20, 2006
Delft University of Technology
ui
xi
v
i
v
wi
w
41
Proof: PC T (7 ) is
p Σ2 -complete t
s u
ui
v
i
v
wi
w
ei f i gi x
September 20, 2006
Delft University of Technology
xi
42
J. Renze Steenhuisen September 20, 2006 1
Delft University of Technology Delft University of Technology
Overview •
•
•
Variant I: Pure Coordination Variant II: Coordinated Assignment Variant III: Complete Coordination
September 20, 2006
Delft University of Technology
2
A Depot Example
A1
A2
t1
t3 d
t4
t2
September 20, 2006
Delft University of Technology
t1
t2
A1
A2
t4
t3
3
A Fixed Task Instance t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 4
Coordination (A Problem) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 5
Pure Coordination Recognition Problem: "Are there still potential directed cycles?" P URE C OORDINATION R ECOGNITION INSTANCE: Given a fixed task instance. QUESTION: Is this instance coordinated?
September 20, 2006
Delft University of Technology
6
A Fixed Task Instance t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 7
Coordination (A Solution) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 8
Coordination (A Solution) t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3 9
Pure Coordination Problem: "How can we exclude all possible cycles with a minimal number of additional constraints?" P URE C OORDINATION INSTANCE: Given a fixed task instance and integer K ≥ 0. QUESTION: Can this instance be coordinated by adding at most K arcs?
September 20, 2006
Delft University of Technology
10
Complexity Analysis •
•
P URE C OORDINATION R ECOGNITION CO NP-complete P URE C OORDINATION Σp2 -complete
September 20, 2006
Delft University of Technology
11
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
12
Subclasses of Pure Coordination Limiting the number of tasks per agent PCRT (n)
PCT (n)
n=2
P
NP-complete
n=3
P
NP-complete
n=4
CO NP-complete
Σp2
n=5
CO NP-complete
Σp2
n=6
CO NP-complete
Σp2
n≥7
CO NP-complete
Σp2 -complete
September 20, 2006
Delft University of Technology
13
Subclasses of Pure Coordination Limiting the number of agents PCRA(n)
PCA(n)
n=2
P
NP
n≥3
P
NP-complete
September 20, 2006
Delft University of Technology
14
Proof: PCRA(2) ∈ P t1
t2
A1
A2 t4
September 20, 2006
Delft University of Technology
t3
15
Proof: PCRA(3) ∈ P
A1
t1
t
t6
t3
t5
September 20, 2006
Delft University of Technology
A2
2
t4
A3 16
Proof: 3VC ∝ PCA(3) u
v
z
September 20, 2006
Delft University of Technology
w
y
x
17
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
18
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
19
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
20
Proof: 3VC ∝ PCA(3) u1 u2
v1
w2
v2 w1
y1
z2 z1
September 20, 2006
Delft University of Technology
y2
x2 x1
21
Proof: 3VC ∝ PCA(3) u
v
z
September 20, 2006
Delft University of Technology
w
y
x
22
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
23
The Depot Example
A1
A2
t1
t3 d
t4
t2
September 20, 2006
Delft University of Technology
t1
t2
A1
A2
t4
t3
24
Task Hierarchy Example
bike
t1
t2
OR
AND
car
truck
t 21
•
Tasks: requirements, decomposition
•
Agents: capabilities
September 20, 2006
Delft University of Technology
t 22
25
Coordinated Assignment Recognition Problem: "Is there an assignment without potential directed cycles?" C OORDINATED A SSIGNMENT R ECOGNITION INSTANCE: Given a free task instance. QUESTION: Does there exist a partitioning that is coordinated?
September 20, 2006
Delft University of Technology
26
Coordinated Assignment Problem: "How can we introduce an assignment that is free of potential directed cycles by adding a minimal number of constraints?" C OORDINATED A SSIGNMENT INSTANCE: Given a free task instance and integer K ≥ 0. QUESTION: Does there exist a partitioning that can be coordinated by adding at most K arcs? September 20, 2006
Delft University of Technology
27
Complexity Analysis •
•
C OORDINATED A SSIGNMENT R ECOGNITION p Σ2 -complete C OORDINATED A SSIGNMENT Σp2 -complete
September 20, 2006
Delft University of Technology
28
Polynomial Hierarchy Σ
P −complete 2
P
Σ2
NP NPc
September 20, 2006
Delft University of Technology
coNP P
coNPc
29
Complete Coordination Recognition Problem: "Are all assignments free of potential directed cycles?" C OMPLETE C OORDINATED R ECOGNITION INSTANCE: Given a free task instance. QUESTION: Does it hold for all possible partitions that the resulting fixed task instance is coordinated? September 20, 2006
Delft University of Technology
30
Complete Coordination Problem: "How can we make all assignments free of potential directed cycles by adding a minimal number of constraints?" C OMPLETE C OORDINATION INSTANCE: Given a free task instance and integer K ≥ 0. QUESTION: Can all possible partitions be coordinated by adding at most K arcs? September 20, 2006
Delft University of Technology
31
Complexity Analysis •
•
C OMPLETE C OORDINATION R ECOGNITION p Π2 -complete C OMPLETE C OORDINATION Πp3 -complete
September 20, 2006
Delft University of Technology
32
Polynomial Hierarchy
P
Σ 2 −complete P
Σ2
NP
coNP
NPc
P
P
Π 3 −complete
coNPc
P
Π2 P
Π 2 −complete
September 20, 2006
Delft University of Technology
P
Π3
33
Future Work •
Subclasses of C OORDINATED A SSIGNMENT
•
Subclasses of C OMPLETE C OORDINATION
•
•
Other constraints: • Time windows • Resource constraints Dynamically adding/removing constraints/tasks (i.e., towards coordination during planning)
September 20, 2006
Delft University of Technology
34
Framework and Complexity Results for Coordinating Non-Cooperative Planning Agents
J. Renze Steenhuisen September 20, 2006 35
Delft University of Technology Delft University of Technology
Proof: PC T (2 ) is NP-complete u
v
x
w
y
September 20, 2006
Delft University of Technology
z
36
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
37
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
38
Proof: PC T (2 ) is NP-complete u1 u2 w2
x1
w1 y1 y2
September 20, 2006
Delft University of Technology
v2 v1
x2 z2 z1
39
Proof: PC T (2 ) is NP-complete u
v
x
w
y
September 20, 2006
Delft University of Technology
z
40
Proof: PCR T (4 ) is
CO NP-complete t
s u
x
September 20, 2006
Delft University of Technology
ui
xi
v
i
v
wi
w
41
Proof: PC T (7 ) is
p Σ2 -complete t
s u
ui
v
i
v
wi
w
ei f i gi x
September 20, 2006
Delft University of Technology
xi
42
