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

1. 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. 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 (Rothemund et al 04) Autonomous DNA walker (Yin et al 04) 2 Motivation: Self-Assembly

3. 3 Motivation: Complexity Theoretical Study of Self-Assembly How complex?

4. Hamiltonian Path ? How many H Paths? …… Sorting PSPACE 11, 3, 10, 25, 6 Who wins? Playing GO PSPACE-Complete 3 < 6 < 10 < 11 < 25 #P Counting Hamiltonian Path #P- Complete NP Hamiltonian Path NP-Complete P Sorting 4 Complexity 101 Complexity Hierarchy

5. Complexity Hierarchy Self-Assembly Model, Problems …… ? PSPACE PSPACE-Complete Formalize #P #P-Complete NP NP-Complete P 5 Motivation: Complexity Theoretical Study of Self-Assembly

6. AGAP-PAGAP-#AGAP-DGAP 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 6 Roadmap Complexity Hierarchy …… PSPACE PSPACE-Complete #P #P-Complete NP NP-Complete P

7. AGAP-PAGAP-#AGAP-DGAP 7 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

8. AGAP-PAGAP-#AGAP-DGAP Seed vertex Temperature Graph Weight function Temperature: τ = 2 Seed vertex 8 Accretive Graph Assembly System Sequentially constructible?

9. AGAP-PAGAP-#AGAP-DGAP 9 Example: An assembly ordering Assembly Ordering Temperature =2 Seed vertex

10. AGAP-PAGAP-#AGAP-DGAP 10 Example Temperature = 2 Stuck!

11. AGAP-PAGAP-#AGAP-DGAP 11 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

12. AGAP-PAGAP-#AGAP-DGAP Hamiltonian Path ? 12 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

13. AGAP-PAGAP-#AGAP-DGAP • Restricted 3SAT: each variable appears ≤ 3, literal ≤ 2 Top v. Literal v. Bottom v. 13 AGAP is NP-complete • AGAP is in NP • AGAP is NP-hard, using reduction from 3SAT

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

15. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 15 AGAP is NP-complete Proposition: φ is satisfiable⇔ exists an assembly ordering Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

16. AGAP-PAGAP-#AGAP-DGAP 2 2 Stage 1 2 2 2 2 2 2 2 2 2 Stage 2 2 2 2 Stage 4 2 2 2 2 2 2 Stage 3 16 AGAP is NP-complete φ is satisfiable ⇒ exists an assembly ordering T T T FF T T F T Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

17. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 F F F -1 2 2 2 2 2 2 2 2 2 17 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 -1 -1 -1 -1 -1 -1 Temperature = 2

18. AGAP-PAGAP-#AGAP-DGAP 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 18 AGAP is NP-complete • Theorem: AGAP is NP-complete Seed vertex -1 -1 -1 -1 -1 -1 Temperature = 2

19. AGAP-PAGAP-#AGAP-DGAP Hamiltonian Path ? 19 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

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

21. AGAP-PAGAP-#AGAP-DGAP How many H Paths? 21 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is • #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

22. AGAP-PAGAP-#AGAP-DGAP 22 #AGAP is #P-complete • Parsimonious reduction from PERMANENT, i.e., counting number of perfect matchings in a bipartite graph Reduction gadget PERMANENT

23. AGAP-PAGAP-#AGAP-DGAP temperature = 2 b 2 -2 c a 6 23 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

24. AGAP-PAGAP-#AGAP-DGAP Nature: e.g. programmed cell death Programmed cell death (NASA) • Engineering: • e.g. remove scaffolds Tower Scaffold 24 Self-Destructible System

25. 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 25 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.

26. AGAP-PAGAP-#AGAP-DGAP Playing GO 26 Roadmap Complexity Hierarchy Accretive Graph Assembly Problem …… • AGAP is NP-complete • Planar AGAP is NP-complete • #AGAP/Stochastic AGAP is #P-complete PSPACE PSPACE-Complete Self-Destructible Graph Assembly Problem #P • DGAP is PSPCAE-complete #P-Complete NP NP-Complete P

27. AGAP-PAGAP-#AGAP-DGAP • Classical tiling • TM simulation • (Rothemund & Winfree 00) • Our cyclic • gadget • Integration 27 DGAP is PSPACE complete • DGAP is PSPACE-complete • Reduction from IN-PLACE ACCEPTANCE • Proof Scheme

28. AGAP-PAGAP-#AGAP-DGAP Features • Genaral graph • Repulsion • Self-destructible Related work • Self-assembly of DNA graphs (Jonoska et al 99) Future • “Towards a mathematical theory of self-assembly” • (Adleman99) 28 Conclusion Summary Complexity Hierarchy …… Accretive Graph Assembly Problem • AGAP is NP-complete • Planar AGAP is NP-complete PSPACE • #AGAP/Stochastic AGAP is #P-complete PSPACE-Complete Self-Destructible Graph Assembly Problem • DGAP is PSPCAE-complete #P #P-Complete • Self-assembly using graph grammars (Klavins et al 04) NP • Tiling scheme(Wang61, Rothemund & Winfree00) NP-Complete P