1. Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructible Systems Peng Yin Joint with John H. Reif and Sudheer Sahu 1 Department of Computer Science, Duke University DNA11, June 7 th, 2005

2. Motivation: Self-Assembly 2 Self-Assembly: Small objects autonomously associate into larger complex Scientific importance: Ubiquitous phenomena in nature Crystal saltEukaryotic cell Engineering significance: Powerful nano-scale & meso-scale construction methods Autonomous DNA walker (Yin et al 04) Algorithmic DNA lattice (Rothemund et al 04)

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

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

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

6. Roadmap 6 AGAP-PAGAP-#AGAP-DGAPNP P NP-Complete #P PSPACE #P-Complete Complexity Hierarchy PSPACE- Complete …………………… Accretive Graph Assembly Problem Self-Destructible Graph Assembly Problem AGAP is NP-complete #AGAP/Stochastic AGAP is #P-complete DGAP is PSPCAE-complete Planar AGAP is NP-complete

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

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

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

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

11. Accretive Graph Assembly Problem 11 Graph Seed vertex Weight function Temperature Temperature: τ = 2 Seed vertex 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- AGAP-PAGAP-#AGAP-DGAP

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

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

14. AGAP is NP-complete 14 AGAP is in NP AGAP is NP-hard, using reduction from 3SAT Restricted 3SAT: each variable appears < 3, literal < 2 Seed vertex Literal v. Top v. Bottom v. Temperature = 2 AGAP- AGAP-PAGAP-#AGAP-DGAP 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

15. AGAP is NP-complete 15 Proposition: φ is satisfiable ⇔ exists an assembly ordering Temperature = 2 AGAP- AGAP-PAGAP-#AGAP-DGAP Seed vertex 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

16. AGAP is NP-complete 16 Seed vertex T T T F F T T F T φ is satisfiable ⇒ exists an assembly ordering Stage 1 Stage 2 Stage 3 Stage 4 Temperature = 2 AGAP- AGAP-PAGAP-#AGAP-DGAP 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

17. AGAP is NP-complete 17 φ is satisfiable ⇐ exists an assembly ordering Seed vertex F F F 2 Total support ≤-1+2=1< 2 = temperature! Exists at least one TRUE literal in each clause; proof by contradiction Temperature = 2 AGAP- AGAP-PAGAP-#AGAP-DGAP 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

18. AGAP is NP-complete 18 Theorem: AGAP is NP-complete Seed vertex Temperature = 2 AGAP- AGAP-PAGAP-#AGAP-DGAP 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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

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

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

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

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

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

25. Self-Destructible Graph Assembly System 25 Temperature Slot Graph Slot Graph Vertex set Vertex set Association rule Association rule: M ⊆ S X V Seed 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. Weight func: V(s a ) X V(s b ) → Z, (s a, s b ) ∈ E -DGAP AGAP-PAGAP-#AGAP-DGAP

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

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

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