1
Complexity of Graph Self-Assembly in Accretive
Systems and Self-Destructible Systems
Peng Yin
Joint with
John H. Reif and Sudheer Sahu
Department of Computer Science, Duke University
DNA11, June 7th, 2005
2
Motivation: Self-Assembly
Self-Assembly:
Small objects autonomously associate
into larger complex
Scientific importance:
Ubiquitous phenomena in nature
Crystal salt
Eukaryotic cell
Engineering significance:
Powerful nano-scale & meso-scale
construction methods
Algorithmic
DNA lattice
Autonomous
DNA walker
(Rothemund et al 04)
(Yin et al 04)
3
Motivation: Complexity Theoretical Study of Self-Assembly
How complex?
4
Complexity 101
Complexity
Hierarchy
……
Hamiltonian
?
Who
wins?
How
many
HPath
Paths?
Sorting
11, 3, 10, 25, 6
PSPACE
Playing
GO
PSPACEComplete
#P
Counting
#PComplete Hamiltonian Path
NP
NP-Complete Hamiltonian
Path
P
Sorting
3 < 6 < 10 < 11 < 25
5
Motivation: Complexity Theoretical Study of Self-Assembly
Self-Assembly Model,
Problems
Formalize
?
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
6
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
7
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
8
Accretive Graph Assembly System
Seed
vertex
Graph
Temperature:
τ=2
Seed
vertex
Temperature
Weight
function
Sequentially
constructible?
AGAP-PAGAP-#AGAP-DGAP
Example: An assembly ordering
Assembly Ordering
Temperature =2
Seed
vertex
9
AGAP-PAGAP-#AGAP-DGAP
Example
Temperature = 2
Stuck!
10
AGAP-PAGAP-#AGAP-DGAP
11
Accretive Graph Assembly Problem
Seed
vertex
Graph
Temperature:
τ=2
Seed
vertex
Temperature
Weight
function
Accretive Graph Assembly Problem:
Given an accretive graph assembly system,
determine whether there exists an assembly
ordering to sequentially assemble the given
target graph.
AGAP-PAGAP-#AGAP-DGAP
12
Roadmap
Accretive Graph Assembly Problem
•
AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
Hamiltonian Path ?
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
13
AGAP is NP-complete
• AGAP is in NP
• AGAP is NP-hard, using reduction from 3SAT
• Restricted 3SAT: each variable appears ≤ 3, literal ≤ 2
Top v.
Literal v.
Bottom v.
AGAP-PAGAP-#AGAP-DGAP
14
AGAP is NP-complete
• AGAP is in NP
• AGAP is NP-hard, using reduction from 3SAT
• Restricted 3SAT: each variable appears < 3, literal < 2
Seed
vertex
2
2
2
2 2
2
2
2
-1
2
2
Top v.
2
-1
Literal v.
-1
-1
2
2
2
Temperature = 2
2
2 2
-1
2
-1
2
2
Bottom v.
AGAP-PAGAP-#AGAP-DGAP
15
AGAP is NP-complete
Proposition: φ is satisfiable ⇔ exists an assembly ordering
Seed
vertex
2
2
2
2 2
2
2
2
-1
-1
2
Temperature = 2
2
-1
-1
2
2
2
2
2
2 2
-1
2
-1
2
2
AGAP-PAGAP-#AGAP-DGAP
16
AGAP is NP-complete
φ is satisfiable ⇒ exists an assembly ordering
T
Seed
vertex
T
T
F
F
T
2
2
2
2
2
-1
Temperature = 2
2
2
Stage 1
2
Stage 2
-1
2
2
2
T
-1
-1
2
F
2
2
2
T
-1
2
2
Stage 4
2
-1
2
2
Stage 3
AGAP-PAGAP-#AGAP-DGAP
17
AGAP is NP-complete
φ is satisfiable ⇐ exists an assembly ordering
Exists at least one TRUE literal in each clause; proof by contradiction
Seed
vertex
Total support ≤-1+2=1< 2 = temperature!
2
2
2
2
2
2
2
22
-1
F
F
-1
Temperature = 2
2
2
2
-1
F
-1
2
2
2
2
-1
2
2
2
-1
2
2
AGAP-PAGAP-#AGAP-DGAP
18
AGAP is NP-complete
•
Theorem: AGAP is NP-complete
Seed
vertex
2
2
2
2
2
2
2
2
-1
-1
2
2
Temperature = 2
2
-1
-1
2
2
2
2
-1
2
2
2
-1
2
2
AGAP-PAGAP-#AGAP-DGAP
19
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
Hamiltonian Path ?
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
20
Planar-AGAP
•
Planar AGAP is NP-complete;reduction from Planar-3SAT
Planar-3SAT
Reduction gadget
Seed
vertex
AGAP-PAGAP-#AGAP-DGAP
21
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is
#P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
How many H Paths?
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
22
#AGAP is #P-complete
• Parsimonious reduction from PERMANENT, i.e., counting number of
perfect matchings in a bipartite graph
PERMANENT
Reduction gadget
AGAP-PAGAP-#AGAP-DGAP
23
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
temperature = 2
NP
b
2
-2
NP-Complete
a
6
c
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
24
Self-Destructible System
Nature:
e.g. programmed cell death
Programmed cell death
(NASA)
Engineering:
e.g. remove scaffolds
Scaffold
Tower
AGAP-PAGAP-#AGAP-DGAP
25
Self-Destructible Graph Assembly System
Slot Graph
Association
rule
Vertex
set
Slot
Graph
Weight
func.
Seed
Temperature
Vertex
set
Seed
Association rule: M ⊆ S X V
Self-Destructible Graph Assembly Problem:
Given a self-destructible graph assembly system,
determine whether there exists a sequence of assembly
operations to sequentially assemble a target graph.
Weight func: V(sa) X V(sb) → Z, (sa, sb) ∈E
AGAP-PAGAP-#AGAP-DGAP
26
Roadmap
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Planar AGAP is NP-complete
• #AGAP/Stochastic AGAP is #P-complete
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
…
…
PSPACE
PSPACEComplete
#P
#P-Complete
Playing GO
NP
NP-Complete
P
Complexity
Hierarchy
AGAP-PAGAP-#AGAP-DGAP
DGAP is PSPACE complete
• DGAP is PSPACE-complete
• Reduction from IN-PLACE ACCEPTANCE
Proof Scheme
Classical tiling
TM simulation
(Rothemund & Winfree 00)
Integration
Our cyclic
gadget
27
AGAP-PAGAP-#AGAP-DGAP
28
Conclusion
Summary
Features
Accretive Graph Assembly Problem
• AGAP is NP-complete
• Genaral graph
• Planar AGAP is NP-complete
• Repulsion
• #AGAP/Stochastic AGAP is #P-complete
• Self-destructible
Self-Destructible Graph Assembly Problem
• DGAP is PSPCAE-complete
Related work
• Self-assembly of DNA graphs (Jonoska et al 99)
• Self-assembly using graph grammars (Klavins et al 04)
…
…
PSPACE
PSPACEComplete
#P
#PComplete
NP
•Tiling scheme (Wang61, Rothemund & Winfree00)
Future
• “Towards a mathematical theory of self-assembly”
(Adleman99)
NPComplete
P
Complexity
Hierarchy