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Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructible Systems Peng Yin

Explore the theoretical study of self-assembly complexity in accretive and self-destructible systems, including previous work, limitations, and detailed models. Discover the scientific and engineering significance of self-assembly phenomena.

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Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructible Systems Peng Yin

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

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

  3. 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

  4. 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

  5. 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

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

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

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

  9. 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

  10. 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

  11. AGAP-PAGAP-#AGAP-DGAP • Restricted 3SAT: each variable appears ≤ 3, literal ≤ 2 Top v. Literal v. Bottom v. 11 AGAP is NP-complete • AGAP is in NP • AGAP is NP-hard, using reduction from 3SAT

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

  13. 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

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

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

  16. AGAP-PAGAP-#AGAP-DGAP 2 F F F -1 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 -1 -1 -1 -1 Temperature = 2

  17. 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

  18. 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

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

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

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

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

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

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

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

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

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

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

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

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

  31. 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

  32. 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

  33. 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

  34. AGAP-PAGAP-#AGAP-DGAP N(S) Hall’s Theorem: |N(S)| < |S| S 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

  35. 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

  36. 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

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

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

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

  40. 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

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

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

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

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

  45. 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

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