John H. Reif and Sudheer Sahu 1 Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructible Systems Peng Yin Joint with John H. Reif and Sudheer Sahu

Motivation: Self-Assembly 2 Motivation: Self-Assembly Self-Assembly: Small objects autonomously associate into larger complex Eukaryotic cell Scientific importance: Ubiquitous phenomena in nature Crystal salt Engineering significance: Powerful nano-scale & meso-scale construction methods DNA walker (Yin et al 04) DNA Lattice (Yan et al 03)

Motivation: Complexity Theoretical Study of Self-Assembly 3 Motivation: Complexity Theoretical Study of Self-Assembly Previous work Wang tiling models (1961) Rothemund & Winfree (2000) Subsequent work Limitations No repulsion modeled Only rectangular grids modeled Our model Repulsion force; graph setting Accretive graph assembly model & Self-Destructible graph assembly model Sequential assembly of a target graph

AGAP-PAGAP-#AGAP-DGAP 4 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 PSPACE-complete

AGAP-PAGAP-AGAP-#AGAP-DGAP 5 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 PSPACE-complete

Sequentially constructible? AGAP-PAGAP-#AGAP-DGAP 6 Accretive Graph Assembly System Seed vertex Temperature Temperature: τ = 2 Graph Weight function Sequentially constructible?

AGAP-PAGAP-#AGAP-DGAP 7 Example: An assembly ordering Assembly Ordering Support ≥temperature Temperature =2

AGAP-PAGAP-#AGAP-DGAP 8 Example Temperature = 2 Stuck!

AGAP-PAGAP-#AGAP-DGAP 9 Accretive Graph Assembly Problem Seed vertex Temperature Graph Weight function Temperature: τ = 2 Accretive Graph Assembly Problem: Given an accretive graph assembly system, determine whether there exists an assembly ordering to sequentially assemble the given target graph. Seed vertex

AGAP-PAGAP-#AGAP-DGAP 10 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 PSPACE-complete

AGAP-PAGAP-#AGAP-DGAP 11 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 12 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 -1 Top v. Literal v. Bottom v. Temperature = 2

Seed 2 vertex -1 -1 -1 -1 -1 -1 AGAP is NP-complete AGAP-PAGAP-#AGAP-DGAP 13 AGAP is NP-complete Proposition: φ is satisfiable ⇔ exists an assembly ordering Seed vertex 2 -1 -1 -1 -1 -1 -1 Temperature = 2

Seed 2 vertex -1 -1 -1 -1 -1 -1 AGAP is NP-complete AGAP-PAGAP-#AGAP-DGAP 14 AGAP is NP-complete φ is satisfiable ⇒ exists an assembly ordering T T T F F T T F T Seed vertex 2 Stage 1 -1 -1 Stage 2 Stage 4 -1 -1 -1 -1 Stage 3 Temperature = 2

Seed 2 vertex -1 -1 -1 -1 -1 -1 AGAP is NP-complete AGAP-PAGAP-#AGAP-DGAP 15 AGAP is NP-complete φ is satisfiable ⇐ exists assembly an ordering Seed vertex 2 -1 -1 F T -1 -1 T -1 -1 Temperature = 2

Seed 2 vertex -1 -1 -1 -1 -1 -1 AGAP is NP-complete AGAP-PAGAP-#AGAP-DGAP 16 AGAP is NP-complete φ is satisfiable ⇐ exists an assembly ordering Exists at least one TRUE literal in each clause; proof by contradiction Total support ≤-1+2=1< 2 = temperature! Seed vertex 2 -1 -1 2 F -1 -1 -1 -1 -1 Temperature = 2

AGAP-PAGAP-#AGAP-DGAP 17 AGAP is NP-complete Theorem: AGAP is NP-complete Corollary: 4-DEGREE AGAP is NP-complete Seed vertex 2 -1 -1 -1 -1 -1 -1 Temperature = 2

AGAP-PAGAP-#AGAP-DGAP 18 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 PSPACE-complete

AGAP-PAGAP-#AGAP-DGAP 19 Planar AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 20 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 21 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 22 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 23 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 24 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT

AGAP-PAGAP-#AGAP-DGAP 25 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT Seed vertex

AGAP-PAGAP-#AGAP-DGAP 26 Planar-AGAP Planar AGAP is NP-complete;reduction from Planar-3SAT Seed vertex

Seed vertex Planar-AGAP AGAP-PAGAP-#AGAP-DGAP 27 Planar-AGAP Proposition: φ is satisfiable ⇔ exists an assembly ordering Seed vertex

Seed vertex Planar-AGAP φ is satisfiable ⇒ exists an assembly ordering AGAP-PAGAP-#AGAP-DGAP 28 Planar-AGAP φ is satisfiable ⇒ exists an assembly ordering T F F T F F T T F T F T Seed vertex

Seed vertex Planar-AGAP φ is satisfiable ⇐ exists assembly an ordering AGAP-PAGAP-#AGAP-DGAP 29 Planar-AGAP φ is satisfiable ⇐ exists assembly an ordering Seed vertex

AGAP-PAGAP-#AGAP-DGAP 30 Planar-AGAP Theorem: PAGAP is NP-complete Corollary: 5-DEGREE PAGAP is NP-complete Seed vertex

Accretive Graph Assembly Problem AGAP-PAGAP-#AGAP-DGAP AGAP-PAGAP-#AGAP-DGAP 31 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 PSPACE-complete

AGAP-PAGAP-#AGAP-DGAP 32 #AGAP is #P-complete #AGAP: assemble a target vertex set Vt ⊆ V #AGAP is #P-complete Reduction from PERMANENT, i.e., counting number of perfect matchings in a bipartite graph

AGAP-PAGAP-#AGAP-DGAP 33 #AGAP is #P-complete Reduction from PERMANET, i.e., counting number of perfect matchings in a bipartite graph Each matching corresponds to a fixed number of assembly orderings No matching corresponds to no assembly ordering

AGAP-PAGAP-#AGAP-DGAP 34 #AGAP is #P-complete Reduction from PERMANET, i.e., counting number of perfect matchings in a bipartite graph Each matching corresponds to a fixed number of assembly orderings No matching corresponds to no assembly ordering N(S) S Hall’s Theorem: |N(S)| < |S|

AGAP-PAGAP-#AGAP-DGAP 35 Stochastic AGAP is #P-complete #AGAP is #P-complete Stochastic AGAP: Given graph G = (V,E); pick any vertex with equal probability at any time; determine probability of assembling target Vt Stochastic AGAP is #P-complete

AGAP-PAGAP-#AGAP-DGAP 36 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 PSPACE-complete

AGAP-PAGAP-#AGAP-DGAP 37 Self-Destructible System Nature: e.g. programmed cell death Programmed cell death (NASA) Scaffold Tower Engineering: e.g. remove scaffolds

AGAP-PAGAP-#AGAP-DGAP 38 Self-Destructible Graph Assembly System Association rule Association rule: M ⊆ S X V Slot Graph Slot Graph Seed Weight func. Weight func: V(sa) X V(sb) → Z, (sa, sb) ∈E Vertex set Temperature 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.

AGAP-PAGAP-#AGAP-DGAP 39 Self-Destructible Graph Assembly Problem: Example Stepping stone Self-Destruction

AGAP-PAGAP-#AGAP-DGAP 40 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 PSPACE-complete

AGAP-PAGAP-#AGAP-DGAP 41 DGAP is PSPACE complete DGAP is PSPACE-complete Reduction from IN-PLACE ACCEPTANCE Proof Scheme Classical tiling TM simulation Our cyclic gadget Integration

AGAP-PAGAP-#AGAP-DGAP 42 DGAP Proof: TM simulation Wang61, Winfree 2000 Our modified scheme

DGAP Proof: Cyclic Gadget 43 DGAP Proof: Cyclic Gadget Comput. vertices: a, b, c Knocker vertices: x, y, z Anchor vertices: x’, y’, z’

AGAP-PAGAP-#AGAP-DGAP 44 DGAP Proof: Integration of TM and Cyclic Gadget

AGAP-PAGAP-#AGAP-DGAP 45 Summary 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 PSPACE-complete