Natural Computing

, Volume 7, Issue 2, pp 183–201

# On the complexity of graph self-assembly in accretive systems

• Stanislav Angelov
• Sanjeev Khanna
• Mirkó Visontai
Article

## Abstract

We study the complexity of the Accretive Graph Assembly Problem ($${\tt{AGAP}}$$). An instance of $${\tt{AGAP}}$$ consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if the graph G can be assembled by a sequence of vertex additions starting from the seed vertex. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature. A vertex can be added when the total weight to its already built neighbors in the graph is at least τ. The assembly process is sequential meaning that only one vertex can be added at a time. Our first result is that $${\tt{AGAP}}$$ is NP-complete even on planar graphs with maximum degree 3 when edges have only two different types of weights. This resolves the complexity of $${\tt{AGAP}}$$ in the sense that the problem is poly-time solvable when either the maximum degree is at most 2 or the number of distinct edge weights is one, and is NP-complete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of $${\tt{AGAP}}$$ on graphs with maximum degree 3 and two distinct weights: w p and w n . We give a simple system of linear constraints on w p , w n , and τ that determines whether the problem is NP-complete or is poly-time solvable. In the process of establishing this dichotomy, we give a poly-time algorithm to solve a non-trivial class of $${\tt{AGAP}}.$$ Finally, we consider the optimization version of $${\tt{AGAP}}$$ where the goal is to assemble a largest-possible induced subgraph of the given input graph. We show that even on graphs that can be assembled and have maximum degree 3, it is NP-hard to assemble a (1/n 1-ε)-fraction of the input graph for any $$\varepsilon > 0;$$ here n denotes the number of vertices in G.

## Keywords

Graph self-assembly Accretive systems Computational complexity

## Notes

### Acknowledgements

This work was supported in part by NSF Career Award CCR-0093117 and by NSF Award CCF-0635084. The authors would like to thank Péter Biró for helpful discussions. We also thank the anonymous reviewers for their valuable suggestions and comments.

## References

1. Adleman LM, Cheng Q, Goel A, Huang MDA (2001) Running time and program size for self-assembled squares. In: Proceedings of the 33th Annual ACM Symposium on Theory of Computing, 740–748Google Scholar
2. Adleman LM, Cheng Q, Goel A, Huang MDA, Kempe D, de Espanées PM, Rothemund PWK (2002) Combinatorial optimization problems in self-assembly. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 23–32Google Scholar
3. Aggarwal G, Goldwasser M, Kao MY, Schweller RT (2004) Complexities for generalized models of self-assembly. In: Proceedings of the 15th annual ACM-SIAM Symposium on Discrete Algorithms, 880–889Google Scholar
4. Barish RD, Rothemund PWK, Winfree E (2005) Two computational primitives for algorithmic self-assembly: copying and counting. Nano Lett 5(12):2586–2592
5. Broersma H, Li X (1997) Spanning trees with many or few colors in edge-colored graphs. Discussiones Mathematicae Graph Theory 17(2):259–269
6. Chelyapov N, Brun Y, Gopalkrishnan M, Reishus D, Shaw B, Adleman LM (2004) DNA triangles and self-assembled hexagonal tilings. J Am Chem Soc 126(43):13924–13925
7. Chen HL, Cheng Q, Goel A, Huang MDA, de Espanés PM (2004) Invadable selfassembly: combining robustness with efficiency. In: Proceedings of the 15th annual ACM-SIAM Symposium on Discrete Algorithms, 890–899Google Scholar
8. Chen HL, Goel A (2004) Error free self-assembly using error prone tiles. In: Proceedings of the 10th International Workshop on DNA Computing, 62–75Google Scholar
9. Cook M, Rothemund PWK, Winfree E (2003) Self-assembled circuit patterns. In: Proceedings of the 9th International Workshop on DNA Based Computers, 91–107Google Scholar
10. Fujibayashi K, Murata S (2004) A method of error suppression for self-assembling DNA tiles. In: Proceedings of the 10th International Workshop on DNA Computing, 113–127Google Scholar
11. He Y, Chen Y, Liu H, Ribbe AE, Mao C (2005) Self-assembly of hexagonal DNA two-dimensional (2D) arrays. J Am Chem Soc 127(35):12202–12203
12. Jonoska N, Karl SA, Saito M (1999) Three dimensional DNA structures in computing. BioSystems 52:143–153
13. Jonoska N, McColm GL (2005) A computational model for self-assembling flexible tiles. In: Proceedings of the 4th International Conference on Unconventional Computation, 142–156Google Scholar
14. Jonoska N, Sa-Ardyen P, Seeman NC (2003) Computation by self-assembly of DNA graphs. Genetic Program Evolvable Machines 4(2):123–137
15. Kao MY, Schweller R (2006) Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th annual ACM-SIAM Symposium on Discrete Algorithms, 571–580Google Scholar
16. Klavins E (2004) Directed self-assembly using graph grammars. In: Proceedings of the 3rd Conference on Foundations of Nanoscience: self-assembled architectures and devicesGoogle Scholar
17. Klavins E, Ghrist R, Lipsky D (2004) Graph grammars for self-assembling robotic systems. In: Proceedings of the IEEE International Conference on Robotics and Automation, vol. 5:5293–5300Google Scholar
18. LaBean TH, Yan H, Kopatsch J, Liu F, Winfree E, Reif JH, Seeman NC (2000) Construction, analysis, ligation, and self-assembly of DNA triple crossover complexes. J Am Chem Soc 122(9):1848–1860
19. Lagoudakis MG, LaBean TH (1999) 2D DNA self-assembly for satisfiability. In: Proceedings of the 5th DIMACS International Meeting on DNA Based Computers, 139–152Google Scholar
20. Lichtenstein D (1982) Planar formulae and their uses. SIAM J Comp 11(2):329–343
21. Malo J, Mitchell JC, Vnien-Bryan C, Harris JR, Wille H, Sherratt DJ, Turberfield AJ (2005) Engineering a 2D protein-DNA crystal. Angewandte Chemie Int Edn 44(20):3057–3061
22. Middleton AA (1999) Computational complexity of determining the barriers to interface motion in random systems. Phys Rev E 59(3):2571–2577
23. Plesník J (1979) The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two. Inf Process Lett 8(4):199–201
24. Reif JH, Sahu S, Yin P (2004) Compact error-resilient computational DNA tiling assemblies. In: Proceedings of the 10th International Workshop on DNA Computing 293–307Google Scholar
25. Reif JH, Sahu S, Yin P (2005) Complexity of graph self-assembly in accretive systems and self-destructible systems. In: Proceedings of the 11th International Meeting on DNA Computing, 101–112Google Scholar
26. Rothemund PWK (2000) Using lateral capillary forces to compute by self-assembly. Proc Nat Acad Sci USA 97(3):984–989
27. Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2(12):2041–2053
28. Rothemund PWK, Winfree E (2000) The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the 32th Annual ACM Symposium on Theory of Computing, 459–468Google Scholar
29. Sa-Ardyen P, Jonoska N, Seeman NC (2004) Self-assembly of irregular graphs whose edges are DNA helix axes. J Am Chem Soc 126(21):6648–6657, ISSN 0002-7863
30. Sa-Ardyen P, Jonoska N, Seeman NC (2003) Self-assembling DNA graphs. Nat Comp 2(4):427–438
31. Sahu S, Yin P, Reif JH (2005) A self-assembly model of DNA tiles with time dependent glue strength. In: Proceedings of the 11th International Meeting on DNA Computing, 113–124Google Scholar
32. Schulman R, Lee S, Papadakis N, Winfree E (2003) One dimensional boundaries for DNA tile self-assembly. In: Proceedings of the 9th International Workshop on DNA Based Computers, 108–126Google Scholar
33. Schulman R, Winfree E (2004) Programmable control of nucleation for algorithmic self-assembly. In: Proceedings of the 10th International Workshop on DNA Computing, 319–328Google Scholar
34. Soloveichik D, Winfree E (2004) Complexity of self-assembled shapes. In: Proceedings of the 10th International Workshop on DNA Computing, 344–354Google Scholar
35. Soloveichik D, Winfree E (2005) Complexity of compact proofreading for selfassembled patterns. In: Proceedings of the 11th International Meeting on DNA Computing, 125–135Google Scholar
36. Wang H (1961) Proving theorems by pattern recognition II. Bell Syst Tech J 40:1–41Google Scholar
37. Winfree E, Bekbolatov R (2003) Proofreading tile sets: error correction for algorithmic self-assembly. In: Proceedings of the 9th International Workshop on DNA Based Computers, 126–144Google Scholar
38. Winfree E, Liu F, Wenzler LA, Seeman NC (1998) Design and self-assembly of two-dimensional DNA crystals. Nature 394:539–544
39. Yan H, LaBean TH, Feng L, Reif JH (2003) Directed nucleation assembly of DNA tile complexes for barcode-patterned lattices. Proc Natl Acad Sci USA 100(14):8103–8108

## Authors and Affiliations

• Stanislav Angelov
• 1
• Sanjeev Khanna
• 1
• Mirkó Visontai
• 1
Email author
1. 1.Department of Computer and Information Science, School of Engineering and Applied SciencesUniversity of PennsylvaniaPhiladelphiaUSA